text:
\section{Introduction} \label{S:introduction}

Graph labelings were introduced by Kotzig and Rosa \cite{Kotzig} in the 1970s.
Various applications to graph labelings include graph decomposition problems, radar pulse code designs, X-ray
crystallography and communication network models.
We refer the interested reader to J. A. Gallian's comprehensive dynamic survey on graph labelings \cite{Gallian}
for further investigation.

The reader should consult Chartrand, Lesniak, and Zhang \cite{Chartrand} for concepts and notation not explicitly defined in this paper.
The graphs in this paper are connected multigraphs.
The concept of a $C_4$-face-magic labeling was first applied to planar graphs.
We apply the concept to graphs embedded on a projective plane.
For a planar (toroidal, Klein bottle, projective) graph $G = (V, E)$ embedded in the plane (torus, Klein bottle, projective plane),
let $\mathcal{F}(G)$ denote the set of faces of $G$.
Then, $G$ is called a \textit{$C_n$-face-magic planar (toroidal, Klein bottle, projective)} graph if there
exists a bijection $f: V(G) \to \{1, 2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong C_n$,
the sum of all the vertex labels aroiund $C_n$ is a constant $S$. Here, the constant $S$ is called
a \textit{$C_n$-face-magic value} of $G$.
More generally, $C_n$-face-magic planar graph labelings are a special case of $(a, b, c)$-magic labeling
introduced by Lih \cite{Lih}.
For assorted values of $a, b$ and $c$, Baca and others \cite{Baca1, Baca2, Baca3, Hsieh, Kasif, Kath, Lih} have analyzed
the problem for various classes of graphs.
Wang \cite{Wang} showed that the toroidal grid graphs $C_m \times C_n$ are antimagic for all integers $m,n\geqslant 3$.
Butt et al. \cite{Buttetal} investigated face antimagic labelings on toroidal and Klein bottle grid graphs.

Curran, Low and Locke \cite{CurranLow, CurranLowLocke1} investigated $C_4$-face-magic labelings
on an $m \times n$ toroidal grid graph.
They showed that $C_m \times C_n$ admits a $C_4$-face-magic labeling if and only if either $m=2$, or $n=2$, or both $m$ and $n$ are even.
Curran, Low and Locke \cite{CurranLowLocke2} also examined $C_4$-face-magic labelings on an $m \times n$
Klein bottle grid graph.
They showed that an $m \times n$ Klein bottle grid graph  admits a $C_4$-face-magic labeling if and only if $n$ is even.
In this paper, we consider $C_4$-face-magic projective labelings on an $m \times n$
projective grid graph.

Curran \cite{Curran2} showed that an $m \times n$ projective grid graph  admits a $C_4$-face-magic labeling if and only if both $m$ and $n$ have the same parity.
When $m$ and $n$ are even, the $C_4$-face-magic value of a $C_4$-face-magic labeling on an $m\times n$ projective grid graph must be $2mn+2$.
Also, when $m$ and $n$ are odd, the $C_4$-face-magic value of a $C_4$-face-magic labeling on an $m\times n$ projective grid graph is
either $2mn+1$, $2mn+2$, or $2mn+3$.
The $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+2$ were characterized in \cite{Curran2}.
In this paper, we determine a category of the $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+1$ or $2mn+3$.
We conjecture that these are the only $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ having  $C_4$-face-magic value $2mn+1$ or $2mn+3$.

\section{Preliminaries} \label{S:preliminaries}

In this section we introduce definitions and known results about $C_4$-face-magic projective labelings on an $m \times n$ projective grid graph.

\begin{definition}
Let $m$ and $n$ be integers such that $m,n\geqslant 2$.
The {\it $m \times n$ projective grid graph}, denoted by $\mathcal{P}_{m,n}$, is the graph whose vertex set is
\begin{align*}
V\left( \mathcal{P}_{m,n} \right)
=\left\{ \left( i,j\right) :1\leqslant i\leqslant m,1\leqslant j\leqslant n\right\},
\end{align*}
and whose edge set consists of the following edges:
\begin{itemize}
\item there is an edge from $(i,j)$ to $(i,j+1)$, for $1\leqslant i \leqslant m$ and $1\leqslant j \leqslant n-1$,
\item there is an edge from $(i,n)$ to $(m+1-i,1)$, for $1\leqslant i \leqslant m$,
\item there is an edge from $(i,j)$ to $(i+1,j)$, for $1\leqslant i \leqslant m-1$ and $1\leqslant j \leqslant n$, and
\item there is an edge from $(m,j)$ to $(1,n+1-j)$, for $1\leqslant j \leqslant n$.
\end{itemize}
The graph $\mathcal{P}_{m,n}$ has a natural embedding on the projective plane.
This graph is a multigraph since there are double edges on the vertex sets $\{ (1,1), (m,n)\}$ and $\{ (m,1), (1,n)\}$.
\end{definition}

\begin{example} \label{exampleFiveByFiveProjectiveGridGraph}
  The $5 \times 5$ projective grid graph $\mathcal{P}_{5,5}$ is illustrated in Fig. \ref{figFiveByFiveProjectiveGridGraph}.
  Due to the orientation of the vertices in $\mathcal{P}_{m,n}$, we refer to the vertices
  $\{ (i,j): 1 \leqslant j \leqslant n\}$ as column $i$ of $V(\mathcal{P}_{m,n})$ and
  $\{ (i,j): 1 \leqslant i \leqslant m\}$ as row $j$ of $V(\mathcal{P}_{m,n})$.
  \begin{figure}
\begin{picture}(225,225)(-25,-25)

\multiput(0,0)(40,0){5}{\circle*{5}}
\multiput(0,40)(40,0){5}{\circle*{5}}
\multiput(0,80)(40,0){5}{\circle*{5}}
\multiput(0,120)(40,0){5}{\circle*{5}}
\multiput(0,160)(40,0){5}{\circle*{5}}

\put(-20,0){\line(1,0){200}}
\put(-20,40){\line(1,0){200}}
\put(-20,80){\line(1,0){200}}
\put(-20,120){\line(1,0){200}}
\put(-20,160){\line(1,0){200}}

\put(0,-20){\line(0,1){200}}
\put(40,-20){\line(0,1){200}}
\put(80,-20){\line(0,1){200}}
\put(120,-20){\line(0,1){200}}
\put(160,-20){\line(0,1){200}}

\put(2,5){\small (1,1)}
\put(42,5){\small (2,1)}
\put(82,5){\small (3,1)}
\put(122,5){\small (4,1)}
\put(162,5){\small (5,1)}

\put(2,45){\small (1,2)}
\put(42,45){\small (2,2)}
\put(82,45){\small (3,2)}
\put(122,45){\small (4,2)}
\put(162,45){\small (5,2)}

\put(2,85){\small (1,3)}
\put(42,85){\small (2,3)}
\put(82,85){\small (3,3)}
\put(122,85){\small (4,3)}
\put(162,85){\small (5,3)}

\put(2,125){\small (1,4)}
\put(42,125){\small (2,4)}
\put(82,125){\small (3,4)}
\put(122,125){\small (4,4)}
\put(162,125){\small (5,4)}

\put(2,165){\small (1,5)}
\put(42,165){\small (2,5)}
\put(82,165){\small (3,5)}
\put(122,165){\small (4,5)}
\put(162,165){\small (5,5)}

\put(-35,158){$e_{1}$}
\put(-35,118){$e_{2}$}
\put(-35,78){$e_{3}$}
\put(-35,38){$e_{4}$}
\put(-35,-2){$e_{5}$}

\put(-3,-30){$e_{6}$}
\put(37,-30){$e_{7}$}
\put(77,-30){$e_{8}$}
\put(117,-30){$e_{9}$}
\put(157,-30){$e_{10}$}

\put(190,158){$e_{5}$}
\put(190,118){$e_{4}$}
\put(190,78){$e_{3}$}
\put(190,38){$e_{2}$}
\put(190,-2){$e_{1}$}

\put(-3,185){$e_{10}$}
\put(37,185){$e_{9}$}
\put(77,185){$e_{8}$}
\put(117,185){$e_{7}$}
\put(157,185){$e_{6}$}
\end{picture}
\caption{$5 \times 5$ projective grid graph $\mathcal{P}_{5,5}$.}
\label{figFiveByFiveProjectiveGridGraph}
\end{figure}

\end{example}

The following result determines when $\mathcal{P}_{m,n}$ admits a $C_4$-face-magic projective labeling.

\begin{theorem}[\cite{Curran2}, Theorem 10] \label{thmProjectiveGridIsC4FaceMaigc}
Let $m$ and $n$ be integers such that $m,n\geqslant 2$. Then $\mathcal{P}_{m,n}$ admits
a $C_4$-face-magic projective labeling if and only if $m$ and $n$ have the same parity.
\end{theorem}

Furthermore, one can determine the possible $C_4$-face-magic values of a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
The next lemma determines the $C_4$-face-magic value of a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ when $m$ and $n$ are even integers.

\begin{lemma}[\cite{Curran2}, Lemma 5] \label{lemmaEvenByEvenProjectiveGridC4FaceValue}
Suppose $m \geqslant 2$ and $n \geqslant 2$ are even integers.
Let $\{ x_{i,j} : (i,j)\in V(\mathcal{P}_{m,n}) \}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $S$.
Then $S= 2mn+2$.
\end{lemma}

In fact, one can determine the digon face sum values on the digon vertex sets $\{ (1,1), (m,n)\}$ and $\{(m1), (1,n)\}$
on $\mathcal{P}_{m,n}$ for odd integers $m$ and $n$.

\begin{lemma}[\cite{Curran2}, Lemma 6] \label{lemmaOddByOddProjectiveGridC4FaceValue}
Let $m\geqslant 3$ and $n\geqslant 3$ be odd integers.
Let $\{ x_{i,j} : (i,j)\in V(\mathcal{P}_{m,n}) \}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $S$
Let $D_1=x_{1,1} + x_{m,n}$ and $D_2=x_{m,1} + x_{1,n}$ be the face sums of the two digons constructed from the pair of vertices at opposite corners of $\mathcal{P}_{m,n}$.
Then either
\begin{enumerate}
  \item $S=2mn+1$ and $D_1=D_2= \frac{3}{2} mn+ \frac{1}{2}$,
  \item $S=2mn+2$ and $D_1=D_2=mn+1$, or
  \item $S=2mn+3$ and $D_1=D_2= \frac{1}{2} mn+ \frac{3}{2}$.
\end{enumerate}
\end{lemma}

\begin{example}
Fig. \ref{figFiveByFiveProjectiveGridGraphLabeling} illustrates a $C_4$-face-magic projective labeling on the $5 \times 5$ projective grid graph $\mathcal{P}_{5,5}$
that has $C_4$-face-magic value 53.
\end{example}

\begin{figure}
\begin{picture}(225,225)(-25,-25)

\multiput(0,0)(40,0){5}{\circle*{5}}
\multiput(0,40)(40,0){5}{\circle*{5}}
\multiput(0,80)(40,0){5}{\circle*{5}}
\multiput(0,120)(40,0){5}{\circle*{5}}
\multiput(0,160)(40,0){5}{\circle*{5}}

\put(-20,0){\line(1,0){200}}
\put(-20,40){\line(1,0){200}}
\put(-20,80){\line(1,0){200}}
\put(-20,120){\line(1,0){200}}
\put(-20,160){\line(1,0){200}}

\put(0,-20){\line(0,1){200}}
\put(40,-20){\line(0,1){200}}
\put(80,-20){\line(0,1){200}}
\put(120,-20){\line(0,1){200}}
\put(160,-20){\line(0,1){200}}

\put(2,5){${\scriptstyle x_{1,1}=1}$}
\put(42,5){${\scriptstyle x_{2,1}=25}$}
\put(82,5){${\scriptstyle x_{3,1}=2}$}
\put(122,5){${\scriptstyle x_{4,1}=24}$}
\put(162,5){${\scriptstyle x_{5,1}=3}$}

\put(2,45){${\scriptstyle x_{1,2}=23}$}
\put(42,45){${\scriptstyle x_{2,2}=4}$}
\put(82,45){${\scriptstyle x_{3,2}=22}$}
\put(122,45){${\scriptstyle x_{4,2}=5}$}
\put(162,45){${\scriptstyle x_{5,2}=21}$}

\put(2,85){${\scriptstyle x_{1,3}=6}$}
\put(42,85){${\scriptstyle x_{2,3}=20}$}
\put(82,85){${\scriptstyle x_{3,3}=7}$}
\put(122,85){${\scriptstyle x_{4,3}=19}$}
\put(162,85){${\scriptstyle x_{5,3}=8}$}

\put(2,125){${\scriptstyle x_{1,4}=18}$}
\put(42,125){${\scriptstyle x_{2,4}=9}$}
\put(82,125){${\scriptstyle x_{3,4}=17}$}
\put(122,125){${\scriptstyle x_{4,4}=10}$}
\put(162,125){${\scriptstyle x_{5,4}=16}$}

\put(2,165){${\scriptstyle x_{1,5}=11}$}
\put(42,165){${\scriptstyle x_{2,5}=15}$}
\put(82,165){${\scriptstyle x_{3,5}=12}$}
\put(122,165){${\scriptstyle x_{4,5}=14}$}
\put(162,165){${\scriptstyle x_{5,5}=13}$}

\put(-35,158){$e_{1}$}
\put(-35,118){$e_{2}$}
\put(-35,78){$e_{3}$}
\put(-35,38){$e_{4}$}
\put(-35,-2){$e_{5}$}

\put(-3,-30){$e_{6}$}
\put(37,-30){$e_{7}$}
\put(77,-30){$e_{8}$}
\put(117,-30){$e_{9}$}
\put(157,-30){$e_{10}$}

\put(190,158){$e_{5}$}
\put(190,118){$e_{4}$}
\put(190,78){$e_{3}$}
\put(190,38){$e_{2}$}
\put(190,-2){$e_{1}$}

\put(-3,185){$e_{10}$}
\put(37,185){$e_{9}$}
\put(77,185){$e_{8}$}
\put(117,185){$e_{7}$}
\put(157,185){$e_{6}$}
\end{picture}
\caption{$C_4$-face-magic projective labeling on $\mathcal{P}_{5,5}$ having $C_4$-face-magic value $53$.}
\label{figFiveByFiveProjectiveGridGraphLabeling}
\end{figure}

For odd integers $m \geqslant 3$ and $n \geqslant 3$, the $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic
value $2mn+2$ were characterized in \cite{Curran2}. The statement of this characterization involves several technical definitions.
So, we refer the reader to \cite{Curran2} for the details of this characterization.
However, it is relatively easy to count the number of distinct $C_4$-face-magic projective
labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+2$.
We need the following definition in order to sate this result.

\begin{definition} \label{defnProjectiveFactorizationSequence}
  Suppose there exists a positive integer $k$ such that one of the two following conditions hold.
  \begin{enumerate}
    \item There are factorizations of $m=m_1 m_2 \ldots m_{k}$ and $n=n_1 n_2 \ldots n_k$, where $m_i >1$ and $n_i>1$ for all $1 \leqslant i \leqslant k$.
    \item There are factorizations of $m=m'_1 m'_2 \ldots m'_{k}m'_{k+1}$ and $n=n'_1 n'_2 \ldots n'_{k}$,
    where $m'_i >1$ for all $1 \leqslant i \leqslant k+1$, and $n'_i>1$ for all $1 \leqslant i \leqslant k$.
  \end{enumerate}
  We say that $(m_1,n_1,m_2,n_2,\ldots,m_k,n_k)$ is an $(m,n)$-\textit{projective factorization sequence of length $2k$}.
  Also, we say $(m'_1,n'_1,m'_2,n'_2,\ldots,m'_{k},n'_{k},m'_{k+1})$ is an $(m,n)$-\textit{projective factorization sequence of length $2k+1$}.
  For convenience, we let $n'_{k+1}=1$ and refer to
  $(m'_1,n'_1,m'_2,\allowbreak n'_2,\allowbreak \ldots,\allowbreak m'_{k+1},\allowbreak n'_{k+1})$ as an $(m,n)$-\textit{projective factorization sequence of length $2k+1$}.
  In addition, we say that $(m_1,n_1,m_2,n_2,\ldots,m_k,n_k)$ and $(m'_1,n'_1,m'_2,n'_2,\ldots,\allowbreak m'_{k+1},\allowbreak n'_{k+1})$ are
  $(m,n)$-\textit{pro\-jec\-tive factorization sequences}.

  Furthermore, we let $\tau(m,n)$ denote the \textit{number of distinct $(m,n)$-pro\-jec\-tive factorization sequences}.
\end{definition}

The next result provides the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ having
$C_4$-face-magic value $2mn+2$ for distinct odd integers $m$ and $n$.

\begin{theorem}[\cite{Curran2}, Theorem 47] \label{thmNumberOfCentrallyBalancedC4FaceMagicLabelingsOnRectangle}
  Let $m \geqslant 3$ and $n \geqslant 3$ be distinct odd integers.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$  having $C_4$-face-magic value $2mn+2$
  (up to symmetries on the projective plane) is
  \begin{equation*}
    \bigl( \tau(m,n) + \tau(n,m) \bigr) \, 2^{m/2 +n/2-3} \, (\tfrac{m-1}{2})! (\tfrac{n-1}{2})!.
  \end{equation*}
\end{theorem}

The number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,m}$ having
$C_4$-face-magic value $2m^2+2$ for an odd integer $m$ is stated below.

\begin{theorem}[\cite{Curran2}, Theorem 48] \label{thmNumberOfCentrallyBalancedC4FaceMagicLabelingsOnSquare}
  Let $m \geqslant 3$ be an odd integer.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,m}$ having $C_4$-face-magic value $2m^2+2$
  (up to symmetries on the projective plane) is
  \begin{equation*}
    \tau(m,m) \, 2^{m-3} \, \bigl( (\tfrac{m-1}{2})! \bigr)^2.
  \end{equation*}
\end{theorem}

We now direct our attention to $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$  with $C_4$-face-magic value $2mn+1$ or $2mn+3$.

\begin{definition} \label{defnOrderComplementLabeling}
  Let $X=\{ x_{i,j} : (i,j)\in V(\mathcal{P}_{m,n})\}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  We define a labeling $Y$ on $\mathcal{P}_{m,n}$ given by
  \begin{equation*}
   y_{i,j} = mn+1 -x_{i,j} \mbox{ \ \ \ for all } (i,j)\in V(\mathcal{P}_{m,n}).
  \end{equation*}
  We say that $Y$ is the \textit{order plus one complement of $X$ labeling} on $\mathcal{P}_{m,n}$.
  We call the transformation $\mathcal{C}(X)=Y$ the \textit{order plus one complement labeling transformation}.
\end{definition}

\begin{proposition} \label{propOrderComplemtLabeling}
  Let $X=\{ x_{i,j} : (i,j)\in V(\mathcal{P}_{m,n})\}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$
  with $C_4$-face-magic value $S$.
  Then the order plus one complement of $X$ labeling $Y=\mathcal{C}(X)$ on $\mathcal{P}_{m,n}$ is a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$
  with $C_4$-face-magic value $4mn+4-S$.
\end{proposition}

\begin{proof}
  Let $(i_1,j_1)$, $(i_2,j_2)$, $(i_3,j_3)$, and $(i_4,j_4)$ be the vertices of any $C_4$-face on $\mathcal{P}_{m,n}$. Then
  \begin{equation*}
    \sum_{k=1}^4 x_{i_k,j_k} = S.
  \end{equation*}
  Thus, the order complement of $X$ labeling $Y=\mathcal{C}(X)$ satisfies
   \begin{equation*}
    \sum_{k=1}^4 y_{i_k,j_k} =   \sum_{k=1}^4 (mn+1 -x_{i_k,j_k}) = 4mn+4-S.
  \end{equation*}
  Since $h:\{1,2,\ldots,mn\}\rightarrow \{1,2,\ldots,mn\}$ defined by $h(x)=mn+1-x$ is a bijection,
  so is $f:V(P_m \times P_n) \rightarrow \{1,2,\ldots,mn\}$ defined by $f(i,j)= h( x_{i,j})$.
\end{proof}

\begin{remark} \label{remOrderComplementLabelingCorrespondence}
  We observe that $\mathcal{C}(\mathcal{C}(X))=X$ for all $C_4$-face-magic projective labelings
  $X$ on $\mathcal{P}_{m,n}$.
  Thus, $\mathcal{C}$ is a one-to-one correspondence between
  $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $S$ and
  $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $4mn+4-S$.

  Hence, for odd integers $m\geqslant 3$ and $n \geqslant 3$,
  $\mathcal{C}$ is a one-to-one correspondence between
  $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $2mn+1$ and
  those with $C_4$-face-magic value $2mn+3$.
\end{remark}

\section{Bicentrally balanced $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$} \label{S:NonstandardSumProjectiveGrid}

\begin{notation}
Throughout this section, we assume that both $m\geqslant 3$ and $n\geqslant 3$ are odd integers.
We write $m=2m_0 +1$ and $n=2n_0 +1$ for positive integers $m_0$ and $n_0$.
For any positive integer $N$, we let $N^{+}=N+1$.
In particular, we have $m_0^{+}=m_0 +1$ and $n_0^{+}=n_0 +1$.
\end{notation}

\begin{notation}
  We refer to the vertex $(\tfrac{1}{2}(m+1),\tfrac{1}{2}(n+1))=(m_0^{+},n_0^{+})$ as the \textit{center} of the projective grid graph $\mathcal{P}_{m,n}$.
  The graph automorphisms of $\mathcal{P}_{m,n}$ that are induced by a homeomorphism of the projective plane are
  described in relation to the center of $\mathcal{P}_{m,n}$.
  We let $R_{\theta}$ denote the rotation by $\theta$ degrees in the counter-clockwise direction about the center.
  The symmetry $H$ ($V$) is the reflection about the center row (column).
  Since the corner vertices of $\mathcal{P}_{m,n}$ are the only vertices incident to a double edge,
  a symmetry of $\mathcal{P}_{m,n}$ sends each corner vertex to another corner vertex.
  Thus, for distinct integers $m$ and $n$, the set of symmetries on $\mathcal{P}_{m,n}$ is $\{ R_{0}, R_{180}, H, V\}$.
  We let $D_{+}$ ($D_{-}$) denote the reflection about the diagonal with positive (negative) slope passing through the center.
  When $m=n$, the set of symmetries on $\mathcal{P}_{m,m}$ is
  $D_4=\{R_{0}, R_{90}, R_{180}, R_{270}, H, V, D_{+}, D_{-}\}$.
\end{notation}

\begin{definition} \label{defnBicentrallyBalanced}
Let $X=\{x_{i,j}: (i,j) \in V(\mathcal{P}_{m,n}) \}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ with
$C_4$-face value $S=2mn+3$.
For all $(i,j) \in V(\mathcal{P}_{m,n})$, let
\begin{align*}
  S(i,j) &= \tfrac{1}{2}mn+ \tfrac{3}{2} \mbox{ if } i+j \mbox{ is even, and } \\
   S(i,j) &= \tfrac{3}{2}mn+ \tfrac{3}{2} \mbox{ if } i+j \mbox{ is odd.}
\end{align*}
We say that $X$ is \emph{bicentrally balanced} if, for all $(i,j) \in V(\mathcal{P}_{m,n})$,
\begin{equation*}
  x_{i,j} + x_{m+1-i,n+1-j} = S(i,j).
\end{equation*}
\end{definition}

\begin{remark} \label{remBicentrallyBalancedHasValue2MNPlus3}
Suppose $X=\{x_{i,j}: (i,j) \in V(\mathcal{P}_{m,n}) \}$ is a $C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ that is bicentrally balanced.
Then the $C_4$-face-magic value $S$ of $X$ is
\begin{equation*}
  S = x_{1,1} + x_{m,n} + x_{2,1} + x_{m-1,n}=  S(1,1) + S(2,1) = 2mn+3.
\end{equation*}
\end{remark}

We observe that the $C_4$-face-magic projective labeling on $\mathcal{P}_{5,5}$ in Fig. \ref{figFiveByFiveProjectiveGridGraphLabeling} is bicentrally balanced.

\begin{lemma} \label{lemmaProjectiveGridSemiC4FaceValueBalance}
Suppose $m \geqslant 3$ and  $n \geqslant 3$ are odd integers.
Let $X = \{ x_{i,j} : (i,j)\in V(\mathcal{P}_{m,n}) \}$ be a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$
with $C_4$-face-magic value $S=2mn+3$.
Then $X$ is bicentrally balanced. Furthermore, we have $x_{m_0^{+},n_0^{+}}= \tfrac{1}{2}S(m_0^{+},n_0^{+})$.
Thus, $x_{m_0^{+},n_0^{+}}=\tfrac{1}{4}mn+ \tfrac{3}{4}$ if $m^{+}_0 + n^{+}_0$ is even, or
$x_{m_0^{+},n_0^{+}}=\tfrac{3}{4}mn+ \tfrac{3}{4}$ if $m^{+}_0 + n^{+}_0$ is odd.
\end{lemma}

\begin{proof}
By Lemma \ref{lemmaOddByOddProjectiveGridC4FaceValue}, the digons formed by the
vertex sets $\{ (1,1), (m,n)\}$ and $\{(m,1),(1,n)\}$ have face values
\begin{align*}
  D_1 &= x_{1,1} + x_{m,n} = \tfrac{1}{2}mn+ \tfrac{3}{2} \mbox{ and } \\
  D_2 &=  x_{m,1} + x_{1,n} = \tfrac{1}{2}mn+ \tfrac{3}{2}, \mbox{ respectively.}
\end{align*}

Suppose that for some integer $1 \leqslant i < m$,
\begin{equation*}
  x_{i,1} + x_{m+1-i,n} =S(i,1).
\end{equation*}
See Definition \ref{defnBicentrallyBalanced} for the definition of $S(i,j)$.
Since
\begin{equation*}
  x_{i,1} + x_{i+1,1} + x_{m+1-i,n}  + x_{m-i,n} =S,
\end{equation*}
we have
\begin{equation*}
  x_{i+1,1} + x_{m-i,n} =S - S(i,1)= S(i+1,1).
\end{equation*}
Hence,
\begin{equation*}
  x_{i,1} + x_{m+1-i,n} =S(i,1)
\end{equation*}
 for all $1 \leqslant i \leqslant m$.
A similar argument shows that
\begin{equation*}
  x_{1,j} + x_{m,n+1-j} =S(1,j)
\end{equation*}
for all $1 \leqslant j \leqslant n$.

We use induction to show that $x_{i,j} + x_{m+1-i,n+1-j} = S(i,j)$ for all $(i,j)\in V\mathcal{P}_{m,n})$.
Suppose there exist integers $1 < i <m$ and $1 < j < n$ such that
\begin{enumerate}
  \item for all $1\leqslant i' <i$ and $1\leqslant j' \leqslant n$, $x_{i',j'} + x_{m+1-i',n+1-j'} = S(i',j')$, and
  \item for all $1\leqslant j' < j$, $x_{i,j'} + x_{m+1-i,n+1-j'} = S(i,j')$.
\end{enumerate}
We need to show that $x_{i,j} + x_{m+1-i,n+1-j} = S(i,j)$.
When we add the two $C_4$-face-values
\begin{align*}
  x_{i-1,j-1} &+ x_{i-1,j} + x_{i,j-1} + x_{i,j} = S \\
  &\mbox{ and } \\
  x_{m+2-i,n+2-j} &+ x_{m+2-i,n+1-j} + x_{m+1-i,n+2-j} + x_{m+1-i,n+1-j} = S,
\end{align*}
we obtain
\begin{align*}
  (x_{i-1,j-1} + x_{m+2-i,n+2-j}) &+ (x_{i-1,j}  + x_{m+2-i,n+1-j} ) \\
  + (x_{i,j-1}+ x_{m+1-i,n+2-j}) &+ (x_{i,j}+ x_{m+1-i,n+1-j}) = 2S.
\end{align*}
Since
\begin{align*}
   x_{i-1,j-1} + x_{m+2-i,n+2-j} &= S(i-1,j-1), \\
   x_{i-1,j}  + x_{m+2-i,n+1-j} &= S(i-1,j), \mbox{ and} \\
   x_{i,j-1}+ x_{m+1-i,n+2-j}  &= S(i,j-1),
\end{align*}
we have
\begin{equation*}
  S(i-1,j-1) + S(i-1,j) + S(i,j-1) + (x_{i,j}+ x_{m+1-i,n+1-j}) = 2S.
\end{equation*}
Thus
\begin{equation*}
x_{i,j} + x_{m+1-i,n+1-j} =S(i,j).
\end{equation*}

Since
\begin{equation*}
  2x_{m_0^{+},n_0^{+}}= x_{m_0^{+},n_0^{+}} + x_{m+1-m_0^{+},n+1-n_0^{+}} = S(m^{+}_0,n^{+}_0),
\end{equation*}
we have
\begin{equation*}
  x_{m_0^{+},n_0^{+}}= \tfrac{1}{2}S(m^{+}_0,n^{+}_0).
\end{equation*}
Thus, $x_{m_0^{+},n_0^{+}}=\tfrac{1}{4}mn+ \tfrac{3}{4}$ if $m^{+}_0 + n^{+}_0$ is even, or
$x_{m_0^{+},n_0^{+}}=\tfrac{3}{4}mn+ \tfrac{3}{4}$ if $m^{+}_0 + n^{+}_0$ is odd.
\end{proof}

\subsection{Structure of a bicentrally balanced $C_4$-face-magic labeling}

\begin{lemma} \label{lemmaStructureOfBicentrallyBalancedLabeling}
   Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
   For $1 \leqslant j \leqslant n_0$, let
   \begin{equation*}
     a_j = x_{1,j} +x_{1,j+1}.
   \end{equation*}
   Then,
   \begin{enumerate}
     \item for all $1 \leqslant i \leqslant m_0$ where $i$ is odd, and $1 \leqslant j \leqslant n_0$, we have
     \begin{align*}
       x_{i,j} + x_{i,j+1} &= a_j,                      &x_{i,n+1-j} + x_{i,n-j} &= S- a_j,  \\
       x_{m+1-i,j} + x_{m+1-i,j+1} &= a_j,  \mbox{and } &x_{m+1-i,n+1-j} + x_{m+1-i,n-j} &= S- a_j, \mbox{\ \ and }
     \end{align*}
     \item for all $1 \leqslant i \leqslant m_0$ where $i$ is even, and $1 \leqslant j \leqslant n_0$, we have
     \begin{align*}
       x_{i,j} + x_{i,j+1} &= S- a_j,                      &x_{i,n+1-j} + x_{i,n-j} &= a_j,  \\
       x_{m+1-i,j} + x_{m+1-i,j+1} &= S-a_j,  \mbox{and } &x_{m+1-i,n+1-j} + x_{m+1-i,n-j} &= a_j.
     \end{align*}
   \end{enumerate}
\end{lemma}

\begin{proof}
  When we equate the two $C_4$-face sums
  \begin{align*}
    x_{i,j} + x_{i,j+1} +  x_{i+1,j} + x_{i+1,j+1}  &= S \mbox{ \ and} \\
     x_{i+1,j} + x_{i+1,j+1} + x_{i+2,j} + x_{i+2,j+1} &=S,
  \end{align*}
  we obtain
  \begin{equation} \label{eqnEveryOtherPair}
     x_{i,j} + x_{i,j+1}  =  x_{i+2,j} + x_{i+2,j+1}.
  \end{equation}
  By (\ref{eqnEveryOtherPair}), for all $1 \leqslant i \leqslant m_0$ where $i$ is odd, and $1\leqslant j \leqslant n_0$, we have
  \begin{equation*}
    x_{i,j} + x_{i,j+1} = a_j \mbox{ \ and \ }  x_{m+1-i,j} + x_{m+1-i,j+1} = a_j.
  \end{equation*}
  Since
  \begin{equation*}
    a_j + x_{2,j} + x_{2,j+1} = x_{1,j} +x_{1,j+1} + x_{2,j} +x_{2,j+1} = S,
  \end{equation*}
  we have
  \begin{equation*}
    x_{2,j} + x_{2,j+1} = S- a_j
  \end{equation*}
  for all $1 \leqslant j \leqslant n_0$.
  Also by (\ref{eqnEveryOtherPair}), for all $1 \leqslant i \leqslant m_0$ where $i$ is even, and $1\leqslant j \leqslant n_0$, we have
  \begin{equation*}
    x_{i,j} + x_{i,j+1} = S-a_j \mbox{ \ and \ }  x_{m+1-i,j} + x_{m+1-i,j+1} = S-a_j.
  \end{equation*}

  Since
  \begin{equation*}
    a_j + x_{1,n+1-j} + x_{1,n-j} = x_{m,j} +x_{m,j+1} + x_{1,n+1-j} +x_{1,n-j} = S,
  \end{equation*}
  we have
  \begin{equation*}
    x_{1,n+1-j} +x_{1,n-j} = S- a_j
  \end{equation*}
  for all $1 \leqslant j \leqslant n_0$.
  Thus, by (\ref{eqnEveryOtherPair}), for all $1 \leqslant i \leqslant m_0$ where $i$ is odd, and $1\leqslant j \leqslant n_0$, we have
  \begin{equation*}
    x_{i,n+1-j} + x_{i,n-j} = S - a_j \mbox{ \ and \ }  x_{m+1-i,n+1-j} + x_{m+1-i,n-j} = S - a_j.
  \end{equation*}
  Since
  \begin{equation*}
    (S -a_j) + x_{2,n+1-j} + x_{2,n-j} =  x_{1,n+1-j} +x_{1,n-j} + x_{2,n+1-j} + x_{2,n-j}  = S,
  \end{equation*}
  we have
  \begin{equation*}
    x_{2,n+1-j} + x_{2,n-j}  = a_j
  \end{equation*}
  for all $1 \leqslant j \leqslant n_0$.
  Hence, by (\ref{eqnEveryOtherPair}), for all $1 \leqslant i \leqslant m_0$ where $i$ is even, and $1\leqslant j \leqslant n_0$, we have
  \begin{equation*}
    x_{i,n+1-j} + x_{i,n-j} = a_j \mbox{ \ and \ }  x_{m+1-i,n+1-j} + x_{m+1-i,n-j} = a_j.
  \end{equation*}
\end{proof}

\subsection{Row and column permutations on a bicentrally balanced labeling}

\begin{definition} \label{defnElementaryProjectivePermutationColumns}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Let $\eta$ be a permutation on the set $\{1,2,\ldots,m_0\}$ such that $\eta(i)\equiv i \pmod{2}$ for all $1 \leqslant i \leqslant m_0$.
  We define a labeling on $\mathcal{P}_{m,n}$, $Z=\{z_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$,
  such that for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n$, we have
    \begin{align*}
      z_{i,j} &= x_{\eta(i),j},  \\
      z_{m_0^{+},j} &= x_{m_0^{+},j}, \mbox{ and}\\
       z_{m+1-i,j} &= x_{m+1-\eta(i),j}.
    \end{align*}
    We let $\mathcal{E}_{\eta}$ denote the labeling operation given by $\mathcal{E}_{\eta}(X)=Z$.
\end{definition}

\begin{lemma}  \label{lemmaElementaryProjectivePermutationColumns}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$, and
  let $\eta$ be a permutation on the set $\{1,2,\ldots,m_0\}$ such that $\eta(i)\equiv i \pmod{2}$ for all $1 \leqslant i \leqslant m_0$.
  Let $\mathcal{E}_{\eta}$ be the labeling operation defined in Definition \ref{defnElementaryProjectivePermutationColumns}.
  Then the labeling $Z=\mathcal{E}_{\eta}(X)$ is a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
\end{lemma}

\begin{proof}
 We first verify that $Z$ is bicentrally balanced.
 Suppose that $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n$.
 Since $\eta(i)-i$ is even, we have
 \begin{equation*}
   z_{i,j} + z_{m+1-i,n+1-j} = x_{\eta(i),j} + x_{m+1-\eta(i),n+1-j}  =S(i,j).
 \end{equation*}
 Furthermore, we have
  \begin{equation*}
   z_{m_0^{+},j} + z_{m+1-m_0^{+},n+1-j} = x_{m_0^{+},j} + x_{m+1-m_0^{+},n+1-j}  = S(m^{+}_0,j).
 \end{equation*}

  Next, we show that $Z$ is a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  For all $1 \leqslant i <m$ and $1 \leqslant j <n$, by Lemma \ref{lemmaStructureOfBicentrallyBalancedLabeling}, we have
  \begin{equation*}
    z_{i,j} + z_{i,j+1} = x_{i,j} + x_{i,j+1} \mbox{ \ and \ }  z_{i+1,j} + z_{i+1,j+1} = x_{i+1,j} + x_{i+1,j+1}.
  \end{equation*}
  Thus
  \begin{align*}
    z_{i,j} &+ z_{i,j+1} + z_{i+1,j} + z_{i+1,j+1} \\
    & = x_{i,j} + x_{i,j+1} + x_{i+1,j} + x_{i+1,j+1} =S.
  \end{align*}
  Since $Z$ is bicentrally balanced, for $1 \leqslant i <m$, we have
  \begin{equation*}
    z_{i,n} +z_{m+1-i,1} + z_{i+1,n} + z_{m-i,1} = S(i,n) + S(i+1,n) = S.
  \end{equation*}
  Also, since $Z$ is bicentrally balanced, for $1 \leqslant j < n$, we have
  \begin{equation*}
    z_{m,j} +z_{1,n+1-j} + z_{m,j+1} + z_{1,n-j} = S(m,j) + S(m,j+1) = S.
  \end{equation*}
\end{proof}

\begin{definition} \label{defnElementaryProjectivePermutationRows}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Let $\kappa$ be a permutation on the set $\{1,2,\ldots,n_0\}$ such that $\kappa(j)\equiv j \pmod{2}$ for all $1 \leqslant j \leqslant n_0$.
  We define a labeling on $\mathcal{P}_{m,n}$, $Z=\{z_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$,
  such that for all $1 \leqslant i \leqslant m$ and $1 \leqslant j \leqslant n_0$, we have
    \begin{align*}
      z_{i,j} &= x_{i,\kappa(j)},  \\
      z_{i,n_0^{+}} &= x_{i,n_0^{+}}, \mbox{ and } \\
      z_{i,n+1-j} &= x_{i,n+1-\kappa(j)}.
    \end{align*}
    We let $\mathcal{E}_{\kappa}$ denote the labeling operation given by $\mathcal{E}_{\kappa}(X)=Z$.
\end{definition}

\begin{lemma}  \label{lemmaElementaryProjectivePermutationRows}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$, and
  let $\kappa$ be a permutation on the set $\{1,2,\ldots,n_0\}$ such that $\kappa(j)\equiv j \pmod{2}$ for all $1 \leqslant j \leqslant n_0$.
  Let $\mathcal{E}_{\kappa}$ be the labeling operation defined in Definition \ref{defnElementaryProjectivePermutationRows}.
  Then the labeling $Z=\mathcal{E}_{\kappa}(X)$ is a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
\end{lemma}

The proof of Lemma \ref{lemmaElementaryProjectivePermutationRows} is similar to the proof of Lemma \ref{lemmaElementaryProjectivePermutationColumns}.

\begin{definition} \label{defnElementaryProjectiveSwapColumns}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Let $\alpha: \{1,2,\ldots,m_0\}\rightarrow \{0,1\}$.
  We define a labeling on $\mathcal{P}_{m,n}$, $Z=\{z_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$,
  such that for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n$, we have
    \begin{align*}
      z_{i,j} &= x_{(1-\alpha(i))i +\alpha(i)(m+1-i),j}, \mbox{ \ \ and} \\
      z_{m+1-i,j} &= x_{\alpha(i)i + (1- \alpha(i)) (m+1-i),j}.
    \end{align*}
    We let $\mathcal{E}_{\alpha}$ denote the labeling operation given by $\mathcal{E}_{\alpha}(X)=Z$.
    The labeling operation $\mathcal{E}_{\alpha}$ has the effect of keeping the labelings on the vertices of columns $i$ and $m+1-i$ the same if $\alpha(i)=0$ and
    swapping the labelings on the vertices of column $i$ with those of column $m+1-i$ if $\alpha(i)=1$.
\end{definition}

\begin{lemma}  \label{lemmaElementaryProjectiveSwapColumns}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$, and
  let $\alpha: \{1,2,\ldots,m_0\}\rightarrow \{0,1\}$.
  Let $\mathcal{E}_{\alpha}$ be the labeling operation defined in Definition \ref{defnElementaryProjectiveSwapColumns}.
  Then the labeling $Z=\mathcal{E}_{\alpha}(X)$ is a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
\end{lemma}

\begin{proof}
  First, we show that $Z$ is bicentrally balanced.
  Suppose $\alpha(i)=0$. Then
  \begin{equation*}
    z_{i,j} = x_{i,j} \mbox{ \ \ and \ \ } z_{m+1-i,j} = x_{m+1-i,j}.
  \end{equation*}
  Thus
  \begin{equation*}
    z_{i,j} +  z_{m+1-i,n+1-j} =  x_{i,j} +  x_{m+1-i,n+1-j} = S(i,j).
  \end{equation*}
  Suppose $\alpha(i)=1$. Then
  \begin{equation*}
    z_{i,j} = x_{m+1-i,j} \mbox{ \ \ and \ \ } z_{m+1-i,j} = x_{i,j}.
  \end{equation*}
  Thus
  \begin{equation*}
    z_{i,j} +  z_{m+1-i,n+1-j} =  x_{m+1-i,j} +  x_{i,n+1-j} = S(m+1-i,j) =S(i,j)
  \end{equation*}
  since $m+1-i \equiv i \pmod{2}$.
  The proof that $Z$ is a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $2mn+3$ is similar to
  that in the proof of Lemma \ref{lemmaElementaryProjectivePermutationColumns}.
\end{proof}

\begin{definition} \label{defnElementaryProjectiveSwapRows}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Let $\delta: \{1,2,\ldots,n_0\}\rightarrow \{0,1\}$.
  We define a labeling on $\mathcal{P}_{m,n}$, $Z=\{z_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$,
  such that for all $1 \leqslant i \leqslant m$ and $1 \leqslant j \leqslant n_0$, we have
    \begin{align*}
      z_{i,j} &= x_{i, (1-\delta(j))j +\delta(j)(n+1-j)}, \mbox{ \ \ and} \\
      z_{i,n+1-j} &= x_{i,\delta(j)j + (1- \delta(j)) (n+1-j)}.
    \end{align*}
    We let $\mathcal{E}_{\delta}$ denote the labeling operation given by $\mathcal{E}_{\delta}(X)=Z$.
    The labeling operation $\mathcal{E}_{\delta}$ has the effect of keeping the labelings on the vertices of rows $j$ and $n+1-j$ the same if $\delta(j)=0$ and
    swapping the labelings on the vertices of row $j$ with those of row $n+1-j$ if $\delta(j)=1$.
\end{definition}

\begin{lemma}  \label{lemmaElementaryProjectiveSwapRows}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$, and
  let $\delta: \{1,2,\ldots,n_0\}\rightarrow \{0,1\}$.
  Let $\mathcal{E}_{\delta}$ be the labeling operation defined in Definition \ref{defnElementaryProjectiveSwapRows}.
  Then the labeling $Z=\mathcal{E}_{\delta}(X)$ is a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
\end{lemma}

The proof of Lemma \ref{lemmaElementaryProjectiveSwapRows} is similar to the proof of Lemma \ref{lemmaElementaryProjectiveSwapColumns}.

\begin{definition}
  We call each of the labeling operations $\mathcal{E}_{\eta}$ in Definition \ref{defnElementaryProjectivePermutationColumns},
  $\mathcal{E}_{\kappa}$ in Definition \ref{defnElementaryProjectivePermutationRows},
  $\mathcal{E}_{\alpha}$ in Definition \ref{defnElementaryProjectiveSwapColumns}, and
  $\mathcal{E}_{\delta}$ in Definition \ref{defnElementaryProjectiveSwapRows} an
  \textit{elementary projective labeling operation}.
\end{definition}

\begin{definition}
  We say that two bicentrally balanced $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ are
  \textit{projective labeling equivalent} if one labeling can be obtained from the other by applying a sequence
  of elementary projective labeling operations to it.
\end{definition}

\subsection{Standard bicentrally balanced labeling}

Given a bicentrally balanced $C_4$-face-magic projective labeling $X$ on $\mathcal{P}_{m,n}$, the next theorem identifies
a canonical bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ that is
projective labeling equivalent to $X$.

\begin{theorem} \label{thmStandardProjectiveLabeling}
  Let $X=\{x_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$ be a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Then there is a unique bicentrally balanced $C_4$-face-magic projective labeling $Z=\{z_{i,j}:(i,j)\in V(\mathcal{P}_{m,n})\}$
  on $\mathcal{P}_{m,n}$ that is projective labeling equivalent to $X$ such that
  \begin{itemize}
    \item $z_{i,n_0^{+}} <  z_{i+2,n_0^{+}}$ for all $1 \leqslant i \leqslant m-2$ and $i+ n^{+}_0$ is even,
    \item $z_{i,n_0^{+}} >  z_{i+2,n_0^{+}}$ for all $1 \leqslant i \leqslant m-2$ and $i+ n^{+}_0$ is odd,
    \item $z_{m_0^{+},j} <  z_{m_0^{+},j+2}$ for all $1 \leqslant j \leqslant n-2$ and $m^{+}_0 +j$ is even, and
    \item $z_{m_0^{+},j} >  z_{m_0^{+},j+2}$ for all $1 \leqslant j \leqslant n-2$ and $m^{+}_0 +j$ is odd.
  \end{itemize}
\end{theorem}

\begin{proof}
  By Lemma \ref{lemmaProjectiveGridSemiC4FaceValueBalance}, we have $x_{m_0^{+},n_0^{+}}=\tfrac{1}{2} S(m^{+}_0,n^{+}_0)$.
  It is easy to check that this value remains the same regardless of the elementary projective labeling operation that we apply to $X$.
  Since $X$ is bicentrally balanced, for all $1 \leqslant i \leqslant m_0$, we have
  \begin{equation*}
    x_{i,n_0^{+}} + x_{m+1-i,n_0^{+}} =S(i,n^{+}_0).
  \end{equation*}
  Thus, either  $x_{i,n_0^{+}} < \tfrac{1}{2}S(i,n^{+}_0)$ or $x_{m+1-i,n_0^{+}} < \tfrac{1}{2}S(i,n^{+}_0)$.
  We define a function $\alpha:\{1,2,\allowbreak \ldots,\allowbreak m_0\}\allowbreak \rightarrow \allowbreak \{0,1\}$ as follows.
  For each $1 \leqslant i \leqslant m_0$, let
  \begin{equation*}
    \alpha(i) = \left\{\begin{array}{ll}
              0, &\mbox{if \ } i+n^{+}_0 \mbox{ is even and } x_{i,n_0^{+}} < \tfrac{1}{2}S(i,n^{+}_0), \\
              1, &\mbox{if \ } i+n^{+}_0 \mbox{ is even and } x_{i,n_0^{+}} > \tfrac{1}{2}S(i,n^{+}_0), \\
              0, &\mbox{if \ } i+n^{+}_0 \mbox{ is odd and } x_{i,n_0^{+}} > \tfrac{1}{2}S(i,n^{+}_0), \mbox{ and } \\
              1, &\mbox{if \ } i+n^{+}_0 \mbox{ is odd and } x_{i,n_0^{+}} < \tfrac{1}{2}S(i,n^{+}_0).
\end{array} \right.
  \end{equation*}
  By Lemma \ref{lemmaElementaryProjectiveSwapColumns},  $\mathcal{E}_{\alpha}(X)$ is a bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Replace $X$ with $\mathcal{E}_{\alpha}(X)$. Then $X$ satisfies,
  for all $1 \leqslant i \leqslant m_0$,
  \begin{align*}
    x_{i,n_0^{+}} &<\tfrac{1}{2}S(i,n^{+}_0) \mbox{ and } x_{m+1-i,n_0^{+}} > \tfrac{1}{2}S(i,n^{+}_0) \mbox{ if } i+n^{+}_0 \mbox{ is even, and } \\
     x_{i,n_0^{+}} &>\tfrac{1}{2}S(i,n^{+}_0) \mbox{ and } x_{m+1-i,n_0^{+}} < \tfrac{1}{2}S(i,n^{+}_0) \mbox{ if } i+n^{+}_0 \mbox{ is odd.}
  \end{align*}
  Choose a permutation $\eta$ of $\{1,2,\ldots,m_0\}$ with $\eta(i)\equiv i \pmod{2}$ for all $1 \leqslant i \leqslant m_0$
  such that for all  $1 \leqslant i \leqslant m_0 -2$,
  \begin{align*}
    x_{\eta(i),n_0^{+}} &< x_{\eta(i+2),n_0^{+}} \mbox{ \ if }  i+n^{+}_0 \mbox{ is even, and } \\
    x_{\eta(i),n_0^{+}} &> x_{\eta(i+2),n_0^{+}} \mbox{ \ if }  i+n^{+}_0 \mbox{ is odd.}
  \end{align*}
  By Lemma \ref{lemmaElementaryProjectivePermutationColumns},
  $\mathcal{E}_{\eta}(X)$ is a  bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Replace $X$ with $\mathcal{E}_{\eta}(X)$. Then $X$ satisfies,
  for all $1 \leqslant i \leqslant m-2$,
  \begin{align*}
    x_{i,n^{+}_0}   &<  x_{i+2,n^{+}_0}  \mbox{ \ if }  i+n^{+}_0 \mbox{ is even, and } \\
     x_{i,n^{+}_0}   &>  x_{i+2,n^{+}_0}  \mbox{ \ if }  i+n^{+}_0 \mbox{ is odd.}
  \end{align*}

  A similar argument allows us to choose a function $\delta:\{1,2,\ldots,n_0\}\rightarrow \{0,1\}$ and a permutation $\kappa$ on $\{1,2,\ldots,n_0\}$
  with $\kappa(j)\equiv j \pmod{2}$ for all $1 \leqslant j \leqslant n_0$
  such that $Z=\mathcal{E}_{\kappa}( \mathcal{E}_{\delta}(X))$
  is a  bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ and
  \begin{align*}
    z_{i,n_0^{+}} &<  z_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is even,}  \\
    z_{i,n_0^{+}} &>  z_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is odd,}  \\
    z_{m_0^{+},j} &<  z_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is even, and} \\
    z_{m_0^{+},j} &>  z_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is odd.}
  \end{align*}
\end{proof}

\begin{definition}
  We refer to the bicentrally balanced $C_4$-face-magic projective labeling $Z$ in Theorem \ref{thmStandardProjectiveLabeling} as the
  \textit{standard projective labeling associated with} $X$.
  We say that $Z$ is a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
\end{definition}

As a result of Theorem \ref{thmStandardProjectiveLabeling}, we only need to find the
standard bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

\begin{example}
Table~\ref{tableProj9x9Labelingexample} illustrates a standard bicentrally balanced $C_4$-face-magic projective
labeling on $\mathcal{P}_{9,9}$.
For convenience, we display the $9 \times 9$ projective grid graph as a $9 \times 9$ checkerboard where each square cell represents a vertex
  and square cells that share an edge are adjacent.
\end{example}

\begin{table}[h]
\hspace{0.0in}\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
31  &51  &32   &47  &36  &46  &40  &42  &41    \\[0.05in]
\hline
53  &30  &52   &34  &48  &35  &44  &39  &43    \\[0.05in]
\hline
28  &54  &29   &50  &33  &49  &37  &45  &38    \\[0.05in]
\hline
65  &18  &64   &22  &60  &23  &56  &27  &55    \\[0.05in]
\hline
16   &66  &17   &62  &21  &61  &25  &57  &26     \\[0.05in]
\hline
68  &15  &67   &19  &63  &20  &59  &24  &58    \\[0.05in]
\hline
4  &78  &5   &74  &9   &73  &13  &69  &14     \\[0.05in]
\hline
80  &3   &79   &7   &75  &8   &71  &12  &70     \\[0.05in]
\hline
1   &81  &2    &77  &6   &76  &10  &72  &11     \\[0.05in]
\hline
\end{tabular}
\caption{A standard bicentrally balanced $C_4$-face magic projective labeling on $\mathcal{P}_{9,9}$ with $C_4$-face-magic value 165.}
\label{tableProj9x9Labelingexample}
\end{table}

\subsection{Partial bicentrally balanced labeling}

We need to introduce labelings on subgrids of $\mathcal{P}_{m,n}$ in order to determine a category of
standard bicentrally balanced $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$.

\begin{definition} \label{defnPartialLabelings}
  Let $m\geqslant 3$ and $n \geqslant 3$ be odd integers, and let $M \geqslant m$ and $N \geqslant n$ be odd integers.
  Let $\mathrm{Grid}(m,n) =\{ (i,j) : 1 \leqslant i \leqslant m \mbox{ and } 1 \leqslant j \leqslant n \}$ be the
  $m \times n$ subgrid of $\mathcal{P}_{M,N}$.
  Let $P_m \times P_n$ represent the $m \times  n$ planar grid subgraph of $\mathcal{P}_{M,N}$ on $\mathrm{Grid}(m,n)$.
  Let $X= \{ x_{i,j} : (i,j)\in \mathrm{Grid}(m,n) \}$ be a labeling on $P_m \times P_n$
  We say $X$ is a \textit{partial bicentrally balanced $C_4$-face-magic labeling on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$} if
  \begin{enumerate}
    \item the sum of the labels on the vertices of each $C_4$-face of $P_m \times P_n$ is $2MN+3$,
    \item $\{ x_{i,j} : (i,j) \in \mathrm{Grid}(m,n) \mbox{ and } i+j \mbox{ is even}\} = \{ 1,2,\ldots,\tfrac{1}{2}mn+\frac{1}{2}\}$,
    \item $\{ x_{i,j} : (i,j) \in \mathrm{Grid}(m,n) \mbox{ and } i+j \mbox{ is odd}\} = \{ MN- \tfrac{1}{2}mn + \tfrac{3}{2}, MN- \tfrac{1}{2}mn + \tfrac{5}{2},\ldots,MN\}$,
    \item $x_{i,j}+ x_{m+1-i,n+1-j} = \tfrac{1}{2} mn+ \tfrac{3}{2}$ if $(i,j)\in\mathrm{Grid}(m,n)$ and $i+j$ is even,
    \item $x_{i,j}+ x_{m+1-i,n+1-j} = 2MN - \tfrac{1}{2} mn + \tfrac{3}{2}$ if $(i,j)\in\mathrm{Grid}(m,n)$ and $i+j$ is odd, and
    \item $X$ satisfies
    \begin{align*}
    x_{i,n_0^{+}} &<  x_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is even,}  \\
    x_{i,n_0^{+}} &>  x_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is odd,}  \\
    x_{m_0^{+},j} &<  x_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is even, and} \\
    x_{m_0^{+},j} &>  x_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is odd.}
  \end{align*}
  \end{enumerate}
\end{definition}

\subsection{Partial alternating lexicographic labeling}

We introduce two partial bicentrally balanced $C_4$-face-magic labelings on an $m \times n$ subgrid of $\mathcal{P}_{M,N}$.

\begin{definition} \label{defnBaseAlternatingLexicographicLabeling}
  Let $m\geqslant 3$ and $n \geqslant 3$ be odd integers, and let $M \geqslant m$ and $N \geqslant n$ be odd integers.
  The \textit{partial horizontal alternating lexicographic labeling on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$},
  denoted by $\mathrm{HALL}_{M,N}(m,n)$, is the labeling
  $\mathrm{HALL}_{M,N}(m,n)=\{x_{i,j} : (i,j)\in \mathrm{Grid}(m,n) \}$ given by
  \begin{itemize}
  \item $x_{2i-1,2j-1} = m(j-1)+i$, for all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n^{+}_0$,
  \item $x_{2i,2j} = m(j-1)+ m^{+}_0 +i$, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n_0$,
  \item $x_{2i,2j-1} = MN+ m(1-j)+ 1 -i$, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n^{+}_0$, and
  \item $x_{2i-1,2j} = MN - mj+ m^{+}_0 +1 -i$, for all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n_0$.
  \end{itemize}

  Similarly, the \textit{partial vertical alternating lexicographic labeling on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$},
  denoted by $\mathrm{VALL}_{M,N}(m,n)$, is the labeling
  $\mathrm{VALL}_{M,N}(m,n)=\{y_{i,j} : (i,j)\in \mathrm{Grid}(m,n) \}$ given by
  \begin{itemize}
  \item $y_{2i-1,2j-1} = n(i-1)+j$, for all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n^{+}_0$,
  \item $y_{2i,2j} = n(i-1)+ n^{+}_0 +j$, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n_0$,
  \item $y_{2i,2j-1} = MN - ni + n^{+}_0 + 1 -j$, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n^{+}_0$, and
  \item $y_{2i-1,2j} = MN + n(1-i) +1 -j$, for all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n_0$.
  \end{itemize}
\end{definition}

\begin{example}
  The labeling in Fig. \ref{figVALLFiveByFiveProjectiveGridGraphLabeling} is the partial vertical
  alternating lexicographic labeling on the $5 \times 5$ subgrid of $\mathcal{P}_{15,5}$ denoted by $\mathrm{VALL}_{15,5}(5,5)$.
\end{example}

\begin{figure}
\begin{picture}(225,180)(-25,-5)

\multiput(0,0)(40,0){5}{\circle*{5}}
\multiput(0,40)(40,0){5}{\circle*{5}}
\multiput(0,80)(40,0){5}{\circle*{5}}
\multiput(0,120)(40,0){5}{\circle*{5}}
\multiput(0,160)(40,0){5}{\circle*{5}}

\put(0,0){\line(1,0){160}}
\put(0,40){\line(1,0){160}}
\put(0,80){\line(1,0){160}}
\put(0,120){\line(1,0){160}}
\put(0,160){\line(1,0){160}}

\put(0,0){\line(0,1){160}}
\put(40,0){\line(0,1){160}}
\put(80,0){\line(0,1){160}}
\put(120,0){\line(0,1){160}}
\put(160,0){\line(0,1){160}}

\put(2,5){${\scriptstyle y_{1,1}=1}$}
\put(42,5){${\scriptstyle y_{2,1}=73}$}
\put(82,5){${\scriptstyle y_{3,1}=6}$}
\put(122,5){${\scriptstyle y_{4,1}=68}$}
\put(162,5){${\scriptstyle y_{5,1}=11}$}

\put(2,45){${\scriptstyle y_{1,2}=75}$}
\put(42,45){${\scriptstyle y_{2,2}=4}$}
\put(82,45){${\scriptstyle y_{3,2}=70}$}
\put(122,45){${\scriptstyle y_{4,2}=9}$}
\put(162,45){${\scriptstyle y_{5,2}=65}$}

\put(2,85){${\scriptstyle y_{1,3}=2}$}
\put(42,85){${\scriptstyle y_{2,3}=72}$}
\put(82,85){${\scriptstyle y_{3,3}=7}$}
\put(122,85){${\scriptstyle y_{4,3}=67}$}
\put(162,85){${\scriptstyle y_{5,3}=12}$}

\put(2,125){${\scriptstyle y_{1,4}=74}$}
\put(42,125){${\scriptstyle y_{2,4}=5}$}
\put(82,125){${\scriptstyle y_{3,4}=69}$}
\put(122,125){${\scriptstyle y_{4,4}=10}$}
\put(162,125){${\scriptstyle y_{5,4}=64}$}

\put(2,165){${\scriptstyle y_{1,5}=3}$}
\put(42,165){${\scriptstyle y_{2,5}=71}$}
\put(82,165){${\scriptstyle y_{3,5}=8}$}
\put(122,165){${\scriptstyle y_{4,5}=66}$}
\put(162,165){${\scriptstyle y_{5,5}=13}$}

\end{picture}
\caption{The partial bicentrally balanced $C_4$-face-magic projective labeling $\mathrm{VALL}_{15,5}(5,5)$.}
\label{figVALLFiveByFiveProjectiveGridGraphLabeling}
\end{figure}

\begin{theorem} \label{thmHALLandVALLProjectiveLabelings}
  Let $m\geqslant 3$ and $n \geqslant 3$ be odd integers, and let $M \geqslant m$ and $N \geqslant n$ be odd integers.
  The horizontal alternating lexicographic labeling $\mathrm{HALL}_{M,N}(m,n)$ and the vertical alternating lexicographic labeling $\mathrm{VALL}_{M,N}(m,n)$
  are partial bicentrally balanced $C_4$-face-magic labelings on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$.
\end{theorem}

\begin{proof}
  We show that $\mathrm{VALL}_{M,N}(m,n)$ is a partial bicentrally balanced $C_4$-face-magic labelings on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$.
  The proof that $\mathrm{HALL}_{M,N}(m,n)$ is a partial bicentrally balanced $C_4$-face-magic labelings on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$ is similar.

  We observe that for the vertices $(i,j)$ where $i+j$ even, we assign the labels $1,2,\ldots,\tfrac{1}{2}mn+\tfrac{1}{2}$ in lexicographic order;
however, for the vertices $(i,j)$ where $i+j$ odd, we assign the labels $MN-(\tfrac{1}{2}mn-\tfrac{3}{2}),MN-(\tfrac{1}{2}mn-\tfrac{5}{2}),\ldots,MN$ in reverse lexicographic order.

We have  $y_{2i-1,2j-1} + y_{2i-1,2j}=MN+1$ for $1 \leqslant i \leqslant m_0+1$ and  $1 \leqslant j \leqslant n_0$.
Also, we have $y_{2i,2j-1} + y_{2i,2j}=MN+2$  for $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n_0$.
Thus, for $1 \leqslant i \leqslant m-1$ and  $1 \leqslant j \leqslant n_0$, we have
\begin{equation*}
  y_{i,2j-1} + y_{i,2j} + y_{i+1,2j-1} + y_{i+1,2j}=2MN+3.
\end{equation*}

Next, we have $y_{2i-1,2j} + y_{2i-1,2j+1}=MN+2$ for $1 \leqslant i \leqslant m_0+1$ and $1 \leqslant j \leqslant n_0$.
Also, we have $y_{2i,2j} + y_{2i,2j+1}=MN+1$ for $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n_0$.
Thus, for $1 \leqslant i \leqslant m-1$ and  $1 \leqslant j \leqslant n_0$, we have
\begin{equation*}
  y_{i,2j} + y_{i,2j+1} + y_{i+1,2j} + y_{i+1,2j+1}=2MN+3.
\end{equation*}

We observe that, for $1\leqslant j \leqslant n_0 +1$, $y_{1,2j-1} +y_{m,n+2-2j}=\tfrac{1}{2}mn+\tfrac{3}{2}$, and,
for $1\leqslant j \leqslant n_0$, $y_{1,2j} +y_{m,n+1-2j}=2MN - \tfrac{1}{2}mn+\tfrac{3}{2}$.
Thus, for $1\leqslant j \leqslant n-1$, we have
\begin{equation*}
  y_{1,j} +y_{m,n+1-j} +  y_{1,j+1} +y_{m,n-j} =2MN+3.
\end{equation*}

Similarly, for $1\leqslant i \leqslant m_0 +1$, $y_{2i-1,1} +y_{m+2-2i,n}=\tfrac{1}{2}mn+\tfrac{3}{2}$, and,
for $1\leqslant i \leqslant m_0$, $y_{2i,1} +y_{m+1-2i,n}=2MN- \tfrac{1}{2}mn+\tfrac{3}{2}$.
Thus, for $1\leqslant i \leqslant m-1$, we have
\begin{equation*}
  y_{i,1} +y_{m+1-i,n} +  y_{i+1,1} +y_{m-i,n} =2MN+3.
\end{equation*}

For all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n^{+}_0$,
\begin{equation*}
  y_{2i-1,2j-1} + y_{m+2-2i,n+2-2j} = \tfrac{1}{2} mn + \tfrac{3}{2}.
\end{equation*}
Also, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n_0$,
\begin{equation*}
  y_{2i,2j} + y_{m+1-2i,n+1-2j} = \tfrac{1}{2} mn + \tfrac{3}{2}.
\end{equation*}
We observe that, for all $1 \leqslant i \leqslant m^{+}_0$ and $1 \leqslant j \leqslant n_0$,
\begin{equation*}
  y_{2i-1,2j} + y_{m+2-2i,n+1-2j} = 2MN -\tfrac{1}{2} mn + \tfrac{3}{2}.
\end{equation*}
Similarly, for all $1 \leqslant i \leqslant m_0$ and $1 \leqslant j \leqslant n^{+}_0$,
\begin{equation*}
  y_{2i,2j-1} + y_{m+1-2i,n+2-2j} = 2MN -\tfrac{1}{2} mn + \tfrac{3}{2}.
\end{equation*}

Suppose $n\equiv 1\, (\mathrm{mod}\, 4)$. Then $n^{+}_0$ is odd, and we let $n'_0$ be the  positive integer such that $n^{+}_0=2n'_0 -1$.
Thus
\begin{align*}
  y_{2i-1,2n'_0 -1} &= n(i-1)+n'_0   \mbox{ \ for all }  1 \leqslant i \leqslant m^{+}_0, \mbox{ and } \\
  y_{2i,2n'_0 -1} &= MN - ni + n'_0    \mbox{ \ for all  } 1 \leqslant i \leqslant m_0.
\end{align*}
Hence,
\begin{align}
  y_{i,n_0^{+}} &<  y_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is even, and} \label{eqnStandardCentralColumnIncrease} \\
  y_{i,n_0^{+}} &>  y_{i+2,n_0^{+}} \mbox{ \ for all } 1 \leqslant i \leqslant m-2 \mbox{ and } i+ n^{+}_0 \mbox{ is odd.} \label{eqnStandardCentralColumnDecrease}
  \end{align}
A similar argument shows that (\ref{eqnStandardCentralColumnIncrease}) and (\ref{eqnStandardCentralColumnDecrease}) hold when $n\equiv 3\, (\mathrm{mod}\, 4)$.
Also, a similar argument shows that
    \begin{align*}
    y_{m_0^{+},j} &<  y_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is even, and} \\
    y_{m_0^{+},j} &>  y_{m_0^{+},j+2} \mbox{ \ for all } 1 \leqslant j \leqslant n-2 \mbox{ and } m^{+}_0 +j \mbox{ is odd.}
  \end{align*}
\end{proof}

\subsection{Alternating connected sums}

\begin{definition} \label{defnHorizontalAlternatingConnectedSum}
  Let $r \geqslant 3$ be an odd integer and let $r_0$ be the positive integer such that $r=2r_0 +1$.
  Let $X$ be a partial  bicentrally balanced $C_4$-face-magic labeling on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$.
  Suppose $r \leqslant M/m$.
  The \textit{$r$-horizontal alternating connected sum of $X$}, denoted by $\mathrm{HASC}^r(X)$,
  is the labeling $Y=\{ y_{i,j} : (i,j) \in \mathrm{Grid}(mr,n) \}$ on $\mathrm{Grid}(mr,n)$ given by,
  for all $(i,j) \in \mathrm{Grid}(m,n)$ and for all integers $0 \leqslant k \leqslant r_0$,
  \begin{itemize}
    \item $ y_{(2k-1)m+i,j} = MN - x_{i,j} +(k-\tfrac{1}{2})(mn) +\tfrac{3}{2}$ if $1 \leqslant k \leqslant r_0$ and $i+j$ is odd,
    \item $ y_{(2k-1)m+i,j} = MN - x_{i,j} -(k-\tfrac{1}{2})(mn) + \tfrac{3}{2}$ if $1 \leqslant k \leqslant r_0$ and $i+j$ is even,
    \item $ y_{(2k)m+i,j} = x_{i,j} -k(mn)$ if $0 \leqslant k \leqslant r_0$ and $i+j$ is odd, and
    \item $ y_{(2k)m+i,j} = x_{i,j} +k(mn)$ if $0 \leqslant k \leqslant r_0$ and $i+j$ is even.
  \end{itemize}

  Suppose $r \leqslant N/n$.
  The \textit{$r$-vertical alternating connected sum of $X$}, denoted by $\mathrm{VASC}^r(X)$,
  is the labeling $Y=\{ y_{i,j} : (i,j) \in \mathrm{Grid}(m,nr) \}$ on $\mathrm{Grid}(m,nr)$ given by,
  for all $(i,j) \in \mathrm{Grid}(m,n)$ and for all integers $0 \leqslant k \leqslant r_0$,
  \begin{itemize}
    \item $ y_{i,(2k-1)n+j} = MN - x_{i,j} +(k-\tfrac{1}{2})(mn) +\tfrac{3}{2}$ if $1 \leqslant k \leqslant r_0$ and $i+j$ is odd,
    \item $ y_{i,(2k-1)n+j} = MN - x_{i,j} -(k-\tfrac{1}{2})(mn)  + \tfrac{3}{2}$ if $1 \leqslant k \leqslant r_0$ and $i+j$ is even,
    \item $ y_{i,(2k)n+j} = x_{i,j} -k(mn)$ if $0 \leqslant k \leqslant r_0$ and $i+j$ is odd, and
    \item $ y_{i,(2k)n+j} = x_{i,j} +k(mn)$ if $0 \leqslant k \leqslant r_0$ and $i+j$ is even.
  \end{itemize}
\end{definition}

\begin{example}
  The 3-horizontal alternating connected sum of $\mathrm{HALL}_{15,5}(5,5)$ is given in Table \ref{tableHorizontalAltConnSumofHALLexample}.
  For convenience, we display the $15 \times 5$ projective grid graph as a $15 \times 5$ checkerboard.
\end{example}

\begin{table}[h]
\hspace{0.0in}\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
11  &65  &12   &64  &13  &53  &24  &52  &25   &51  &36  &40  &37  &39  &38   \\[0.05in]
\hline
68  &9   &67   &10  &66  &21  &55  &22  &54   &23  &43  &34  &42  &35  &41  \\[0.05in]
\hline
6   &70  &7    &69  &8   &58  &19  &57  &20   &56  &31  &45  &32  &44  &33   \\[0.05in]
\hline
73  &4   &72   &5   &71  &16  &60  &17  &59   &18  &48  &29  &47  &30  &46    \\[0.05in]
\hline
1    &75  &2   &74  &3   &63  &14  &62  &15   &61  &26  &50  &27  &49  &28    \\[0.05in]
\hline
\end{tabular}
\caption{A 3-horizontal alternating connected sum of $\mathrm{HALL}_{15,5}(5,5)$ on $\mathcal{P}_{15,5}$ with $C_4$-face-magic value 153.}
\label{tableHorizontalAltConnSumofHALLexample}
\end{table}

\begin{theorem} \label{thmHASCandVASCareBicentrallyBalanced}
  Suppose $X$ is a partial  bicentrally balanced $C_4$-face-magic labeling on the $m \times n$ subgrid of $\mathcal{P}_{M,N}$.
  If $r$ is an odd positive integer such that $mr\leqslant M$, then
  the $r$-horizontal alternating connected sum of $X$, $\mathrm{HASC}^r(X)$, is a partial  bicentrally balanced $C_4$-face-magic labelings on the $mr \times n$ subgrid of $\mathcal{P}_{M,N}$.
  Similarly, if $r$ is an odd positive integer such that $nr\leqslant N$, then
  the $r$-vertical alternating connected sum of $X$, $\mathrm{VASC}^r(X)$, is a partial  bicentrally balanced $C_4$-face-magic labelings on the $m \times nr$ subgrid of $\mathcal{P}_{M,N}$.
\end{theorem}

\begin{proof}
  We show that $\mathrm{HASC}^r(X)$, is a partial  bicentrally balanced $C_4$-face-magic labeling on the $mr \times n$ subgrid of $\mathcal{P}_{M,N}$.
  The proof that $\mathrm{VASC}^r(X)$, is a partial  bicentrally balanced $C_4$-face-magic labeling on the $m \times nr$ subgrid of $\mathcal{P}_{M,N}$ is similar.

  First, when $1\leqslant k \leqslant r_0$, we have
  \begin{align*}
    y_{(2k-1)m+2i-1,j} + y_{(2k-1)m+2i-1,j+1}   &=   2MN+3 -( x_{2i-1,j} + x_{2i-1,j+1} ) \mbox{ and } \\
    y_{(2k-1)m+2i,j} + y_{(2k-1)m+2i,j+1}       &=    2MN+3 -( x_{2i,j} + x_{2i,j+1} ).
  \end{align*}
  Thus, for all $1\leqslant i \leqslant m-1$, $1\leqslant j \leqslant n-1$, and $1\leqslant k \leqslant r_0$, we have
  \begin{equation*}
    y_{(2k-1)m+i,j} + y_{(2k-1)m+i,j+1} + y_{(2k-1)m+i+1,j} + y_{(2k-1)m+i+1,j+1} = 2MN+3.
  \end{equation*}

  Second, when $0\leqslant k \leqslant r_0$, we have
  \begin{align*}
    y_{(2k)m+2i-1,j} + y_{(2k)m+2i-1,j+1}   &=   x_{2i-1,j} + x_{2i-1,j+1}  \mbox{ and } \\
    y_{(2k)m+2i,j} + y_{(2k)m+2i,j+1}       &=    x_{2i,j} + x_{2i,j+1}.
  \end{align*}
  Thus, for all $1\leqslant i \leqslant m-1$, $1\leqslant j \leqslant n-1$, and $0\leqslant k \leqslant r_0$, we have
  \begin{equation*}
    y_{(2k)m+i,j} + y_{(2k)m+i,j+1} + y_{(2k)m+i+1,j} + y_{(2k)m+i+1,j+1} = 2MN+3.
  \end{equation*}

  When we set the $C_4$-face sums below equal,
  \begin{equation*}
    x_{i,j} + x_{i,j+1} + x_{i+1,j} + x_{i+1,j+1} = x_{i+1,j} + x_{i+1,j+1} + x_{i+2,j} + x_{i+2,j+1},
  \end{equation*}
  we obtain
  \begin{equation*}
    x_{i,j} + x_{i,j+1}  =  x_{i+2,j} + x_{i+2,j+1}.
  \end{equation*}
  Thus, for all $1 \leqslant j \leqslant n-1$, we have
  \begin{equation} \label{eqnOneSideSumEqualsOtherSide}
    x_{1,j} + x_{1,j+1}  =  x_{m,j} + x_{m,j+1}.
  \end{equation}

  Third, for all $1 \leqslant j \leqslant n-1$ and $1 \leqslant k \leqslant r_0$, we have
  \begin{equation*}
     y_{(2k)m,j} + y_{(2k)m,j+1}  = 2MN+3 -( x_{m,j} + x_{m,j+1} ).
  \end{equation*}
  Similarly, we have
  \begin{equation*}
     y_{(2k)m+1,j} + y_{(2k)m+1,j+1}  =  x_{1,j} + x_{1,j+1}.
  \end{equation*}
  By (\ref{eqnOneSideSumEqualsOtherSide}), we have
  \begin{equation*}
      y_{(2k)m,j} + y_{(2k)m,j+1} + y_{(2k)m+1,j} + y_{(2k)m+1,j+1}  =  2MN+3.
  \end{equation*}
  A similar argument shows that, for all $1 \leqslant j \leqslant n-1$ and $1 \leqslant k \leqslant r_0$, we have
  \begin{equation*}
      y_{(2k-1)m,j} + y_{(2k-1)m,j+1} + y_{(2k-1)m+1,j} + y_{(2k-1)m+1,j+1}  =  2MN+3.
  \end{equation*}

  From the construction of $Y$, we have
  \begin{align*}
    \{ y_{i,j} : (i,j) \in \mathrm{Grid}(mr,n) \mbox{ and } i+j \mbox{ is even}\} &= \{ 1,2,\ldots,\tfrac{1}{2}mnr+\tfrac{1}{2}\}, \mbox{ and } \\
    \{ y_{i,j} : (i,j) \in \mathrm{Grid}(mr,n) \mbox{ and } i+j \mbox{ is odd}\} &= \{ MN- \tfrac{1}{2}mnr + \tfrac{3}{2}, \\
    &MN- \tfrac{1}{2}mnr + \tfrac{5}{2}, \allowbreak \ldots,MN\}.
  \end{align*}
   If $i+j$ is even, we have
   \begin{align*}
     y_{(2k-1)m+i,j} + y_{(2(r_0-k+1)-1)m +(m+1-i),n+1-j} &= 2MN -\tfrac{1}{2} mnr +\tfrac{3}{2}, \mbox{ \ and } \\
     y_{(2k)m+i,j} + y_{2(r_0-k)m +(m+1-i),n+1-j}  &= \tfrac{1}{2} mnr +\tfrac{3}{2}.
   \end{align*}
   If $i+j$ is odd, we have
   \begin{align*}
     y_{(2k-1)m+i,j} + y_{(2(r_0-k+1)-1)m +(m+1-i),n+1-j}  &= \tfrac{1}{2} mnr +\tfrac{3}{2}, \mbox{ \ and }  \\
      y_{(2k)m+i,j} + y_{2(r_0-k)m +(m+1-i),n+1-j} &= 2MN -\tfrac{1}{2} mnr +\tfrac{3}{2}.
   \end{align*}
   Thus, for all $1 \leqslant i \leqslant mr$ and $1 \leqslant j \leqslant n$,
   \begin{align*}
     y_{i,j} + y_{mr+1-i,n+1-j} &= \tfrac{1}{2} mnr +\tfrac{3}{2}, \mbox{ \ if } i+j \mbox{ is even, and} \\
     y_{i,j} + y_{mr+1-i,n+1-j} &= 2MN -\tfrac{1}{2} mnr +\tfrac{3}{2}, \mbox{ \ if } i+j \mbox{ is odd.}
   \end{align*}

   We have,
   for all $1 \leqslant i \leqslant m$ and $0 \leqslant k \leqslant r_0$,
  \begin{align*}
   y_{(2k-1)m+i,n^{+}_0} &= MN - x_{i,n^{+}_0} +(k-\tfrac{1}{2})(mn) +\tfrac{3}{2} \mbox{ \ if } i+n^{+}_0 \mbox{ is odd and} \\
   y_{(2k)m+i,n^{+}_0} &= x_{i,n^{+}_0} +k(mn) \mbox{ \ if } i+n^{+}_0 \mbox{ is even.}
  \end{align*}
   Since, for all  $1 \leqslant i \leqslant m-2$,
   \begin{align*}
    x_{i,n_0^{+}} &<  x_{i+2,n_0^{+}} \mbox{ \ if } i+ n^{+}_0 \mbox{ is even and }    \\
    x_{i,n_0^{+}} &>  x_{i+2,n_0^{+}} \mbox{ \ if } i+ n^{+}_0 \mbox{ is odd,}
  \end{align*}
   we have
   \begin{align*}
    y_{(2k-1)m+i,n_0^{+}} &<  y_{(2k-1)m+i+2,n_0^{+}} \mbox{ \ if }  (2k-1)m+i+ n^{+}_0 \mbox{ is even and }  \\
    y_{(2k)m+i,n_0^{+}} &<  y_{(2k)m+i+2,n_0^{+}} \mbox{ \ if } (2k)m+i+ n^{+}_0 \mbox{ is even.}
  \end{align*}
   In addition, we have
   \begin{align*}
     (k-1)(mn) +\tfrac{1}{2}mn +\tfrac{3}{2} &\leqslant y_{(2k-1)m+i,n^{+}_0} \leqslant k(mn) \\
     &\mbox{ \ if } (2k-1)m+i+n^{+}_0 \mbox{ is even and } \\
     k(mn) + 1 &\leqslant y_{(2k)m+i,n^{+}_0} \leqslant k(mn) +\tfrac{1}{2}mn +\tfrac{1}{2}  \\
     &\mbox{ \ if } (2k)m+i+n^{+}_0 \mbox{ is even.}
   \end{align*}
   Hence, for all $1 \leqslant i \leqslant mr-2$,
   \begin{equation*}
      y_{i,n_0^{+}} <  y_{i+2,n_0^{+}} \mbox{ \ if } i+ n^{+}_0 \mbox{ is even.}
   \end{equation*}
   A similar argument shows that, for all $1 \leqslant i \leqslant mr-2$,
   \begin{equation*}
      y_{i,n_0^{+}} >  y_{i+2,n_0^{+}} \mbox{ \ if } i+ n^{+}_0 \mbox{ is odd.}
   \end{equation*}

   Let $M^{+}_0$ be the positive integer such that $mr= 2M^{+}_0 -1$.
   Then $M^{+}_0 = m r_0 + m^{+}_0$.
   Suppose $r_0$ is odd. Let $r'_0$ be the positive integer such that $r_0 = 2 r'_0 -1$.
   For all $1 \leqslant j \leqslant n$, we have
   \begin{align*}
      y_{(2r'_0 -1)m+m^{+}_0,j} &= MN - x_{m^{+}_0,j} +(r'_0 -\tfrac{1}{2})(mn)  +\tfrac{3}{2} \mbox{ \ if }  m^{+}_0+j \mbox{ is odd and } \\
      y_{(2r'_0 -1)m+m^{+}_0,j} &= MN - x_{m^{+}_0,j} - (r'_0 -\tfrac{1}{2})(mn)  +\tfrac{3}{2} \mbox{ \ if }  m^{+}_0+j \mbox{ is even.}
   \end{align*}
   Since, for all  $1 \leqslant j \leqslant n-2$,
   \begin{align*}
    x_{m_0^{+},j} &<  x_{m_0^{+},j+2} \mbox{ \ if } m^{+}_0 +j  \mbox{ is even and }  \\
    x_{m_0^{+},j} &>  x_{m_0^{+},j+2} \mbox{ \ if } m^{+}_0 +j  \mbox{ is odd,}
  \end{align*}
   we have
   \begin{align*}
    y_{(2r'_0 -1)m+ m_0^{+},j} &<  y_{(2r'_0 -1)m+ m_0^{+},j+2} \mbox{ \ if }  (2r'_0 -1)m+ m^{+}_0 +j \mbox{ is even and }  \\
    y_{(2r'_0 -1)m+ m_0^{+},j} &>  y_{(2r'_0 -1)m+ m_0^{+},j+2} \mbox{ \ if }  (2r'_0 -1)m+ m^{+}_0 +j \mbox{ is odd.}
  \end{align*}
   Thus, for all $1 \leqslant j \leqslant n-2$, we have
   \begin{align}
    y_{M_0^{+},j} &<  y_{M_0^{+},j+2}   \mbox{ \ if } M^{+}_0 +j \mbox{ is even and } \label{eqnConnectedSumLabelCenterColumnIncrease2}  \\
    y_{M_0^{+},j} &>  y_{M_0^{+},j+2}   \mbox{ \ if } M^{+}_0 +j \mbox{ is odd.} \label{eqnConnectedSumLabelCenterColumnDecrease2}
  \end{align}
   A similar argument shows that (\ref{eqnConnectedSumLabelCenterColumnIncrease2}) and (\ref{eqnConnectedSumLabelCenterColumnDecrease2}) hold when $r_0$ is even.
\end{proof}

\subsection{Labelings associated with a projective factorization sequence}

\begin{definition} \label{defnBicentrallyBalancedLabelingAssociatedWithF}
  Let $m\geqslant 3$ and $n\geqslant 3$ be odd integers.
   \begin{enumerate}
     \item Let $F= (m_i, n_i : 1 \leqslant i \leqslant k)$ be an $(m,n)$-projective factorization sequence.
     See Definition \ref{defnProjectiveFactorizationSequence}.
       Let $X_1 =\mathrm{HALL}(m_1,n_1)$.
       For $2 \leqslant i \leqslant k$, let $Y_i =\mathrm{HACS}^{m_i}(X_{i-1})$ and $X_i =\mathrm{VACS}^{n_i}(Y_{i})$.
       The \textit{horizontal bicentrally balanced labeling associated with} $F$ is denoted by $\mathrm{HBBL}(F)=X_k$.
      \item Let $F'= (n'_i, m'_i : 1 \leqslant i \leqslant k)$ be an $(n,m)$-projective factorization sequence.
       Let $X'_1 =\mathrm{VALL}(m'_1,n'_1)$.
       For $2 \leqslant i \leqslant k$, let $Y'_i =\mathrm{VACS}^{n'_i}(X'_{i-1})$ and $X'_i =\mathrm{HACS}^{m'_i}(Y'_{i})$.
       The \textit{vertical bicentrally balanced labeling associated with} $F'$ is denoted by $\mathrm{VBBL}(F')=X'_k$.
   \end{enumerate}
\end{definition}

\begin{theorem} \label{thmStandardProjectiveC4MagicBicentralConstruction}
  Let $m\geqslant 3$ and $n\geqslant 3$ be odd integers.
   Suppose $X$ is constructed in one of the following two ways.
   \begin{enumerate}
     \item Suppose $X= \mathrm{HBBL}(F)$ for some $(m,n)$-projective factorization sequence
     $F=(m_1,\allowbreak n_1,\allowbreak m_2,\allowbreak n_2,\ldots,m_{k},n_{k})$.
     \item Suppose $X= \mathrm{VBBL}(F')$ for some $(n,m)$-projective factorization sequence
     $F'=(n'_1,\allowbreak m'_1,\allowbreak n'_2,\allowbreak m'_2,\ldots,n'_{k},m'_{k})$.
   \end{enumerate}
    Then $X$ is a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.

   Furthermore, distinct $(m,n)$-projective factorization sequences $F_1$ and $F_2$ give rise
   to distinct standard bicentrally balanced $C_4$-face-magic projective labelings $\mathrm{HBBL} \allowbreak (F_1)$ and $\mathrm{HBBL} \allowbreak (F_2)$ on $\mathcal{P}_{m,n}$.
   Similarly, distinct $(n,m)$-projective factorization sequences $F'_1$ and $F'_2$ give rise
   to distinct standard bicentrally balanced $C_4$-face-magic projective labelings $\mathrm{VBBL}(F'_1)$ and $\mathrm{VBBL}(F'_2)$ on $\mathcal{P}_{m,n}$.
\end{theorem}

\begin{proof}
  We first show that $\mathrm{HBBL}(F)$ is a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  By Theorem \ref{thmHALLandVALLProjectiveLabelings}, $X_1 =\mathrm{HALL}_{m,n}(m_1,n_1)$ is a partial
  bicentrally balanced $C_4$-face-magic labeling on the $m_1 \times n_1$ subgrid of $\mathcal{P}_{m,n}$.
  For $1 \leqslant i \leqslant k$, let
  \begin{align*}
    M_i &= m_1 m_2 \cdots m_i \mbox{ and } \\
    N_i &= n_1 n_2 \cdots n_i.
  \end{align*}
  For some integer $2 \leqslant i \leqslant k$, suppose $X_{i-1}$ is a  partial
  bicentrally balanced $C_4$-face-magic labeling on the $M_{i-1} \times N_{i-1}$ subgrid of $\mathcal{P}_{m,n}$.
  By Theorem \ref{thmHASCandVASCareBicentrallyBalanced}, $Y_i =\mathrm{HACS}^{m_i}(X_{i-1})$
  is a  partial bicentrally balanced $C_4$-face-magic labeling on the $M_{i} \times N_{i-1}$ subgrid of $\mathcal{P}_{m,n}$
  and $X_i =\mathrm{VACS}^{n_i}(Y_{i})$ is a  partial
  bicentrally balanced $C_4$-face-magic labeling on the $M_{i} \times N_{i}$ subgrid of $\mathcal{P}_{m,n}$
  Thus $X_k= \mathrm{HBBL}(F)$ is a  partial
  bicentrally balanced $C_4$-face-magic labeling on the $m \times n$ subgrid of $\mathcal{P}_{m,n}$.
  Hence,  $\mathrm{HBBL}(F)$ is a  standard
  bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.

  A similar argument shows that, for an $(n,m)$-projective factorization sequence $F'$,
  $\mathrm{VBBL}\allowbreak (F')$ is a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.

  Let $F_i= (m_{i,j}, n_{i,j} : 1 \leqslant j \leqslant k_i)$, for $i=1,2$, be distinct $(m,n)$-projective factorization sequences.
  We need to show that $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ are distinct
  standard bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.
  Suppose $m_{1,j'}$ and $m_{2,j'}$ are the first entries in $F_1$ and $F_2$, respectively, such that  $m_{1,j'} \ne m_{2,j'}$.

  First, assume $j'=1$. Without loss of generality, we may assume $m_{1,1} < m_{2,1}$.
  Then the labels on $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ are the same on the $m_{1,1} \times 1$ subgrid of $\mathcal{P}_{m,n}$.
  Let $Z=\{y_{i,1} : 1 \leqslant i \leqslant m_{1,1}\}$ be the common labels of
  $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ on the $m_{1,1} \times 1$ subgrid of $\mathcal{P}_{m,n}$.
  Let $z$ be the smallest positive integer such that $z\in \mathrm{Label}_Z(\mathrm{Grid}(m_{1,1}, 1))$
  and $z+1\notin \mathrm{Label}_Z(\mathrm{Grid}(m_{1,1}, 1))$.
  Let $m'=m_{1,1}$.
  Then $z=y_{m',1}=\tfrac{1}{2}(m'+1)$.
  In $\mathrm{HBBL}(F_1)$ we have $y_{2,2}=z+1$, and
  in $\mathrm{HBBL}(F_2)$ we have $y_{m' +2,1}=z+1$.
  Thus $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ are distinct standard bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

  Suppose $j'>1$.
  Without loss of generality, we may assume $m_{1,j'} < m_{2,j'}$.
  Let $m'=m_{1,j'} M_{j' -1}$ and $n'=N_{j' -1}$.
  Let $Z= \{ y_{i,j} : (i,j) \in \mathrm{Grid}(m',n')\}$ be the common labels of
  $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ on the $m' \times n'$ subgrid of $\mathcal{P}_{m,n}$.
  Let $z$ be the smallest positive integer such that $z\in \mathrm{Label}_{Z}(\mathrm{Grid}(m',n'))$
  and $z+1\notin \mathrm{Label}_{Z}(\mathrm{Grid}(m',n'))$.
  Then $z=y_{m',n'}=\tfrac{1}{2}(m' n'+1)$.
  In $\mathrm{HBBL}(F_1)$ we have $y_{2,n' +1}=z+1$ , and
  in $\mathrm{HBBL}(F_2)$ we have $y_{m' +2,1}=z+1$.
  Thus $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ are distinct standard bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

  Suppose $n_{1,j'}$ and $n_{2,j'}$ are the first entries in $F_1$ and $F_2$, respectively, such that  $n_{1,j'} \ne n_{2,j'}$.
  A similar argument to the one above shows that $\mathrm{HBBL}(F_1)$ and $\mathrm{HBBL}(F_2)$ are distinct
  standard bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

  Similarly, we can show that distinct $(n,m)$-projective factorization sequences $F'_1$ and $F'_2$ give rise
   to distinct standard bicentrally balanced $C_4$-face-magic projective labelings $\mathrm{VBBL}(F'_1)$ and $\mathrm{VBBL}(F'_2)$ on $\mathcal{P}_{m,n}$.
\end{proof}

\section{Enumerating bicentrally balanced labelings}

We will enumerate the minimum number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

\begin{notation}
  Let $m\geqslant 3$ be an odd integer.
  We define the function $\beta$ given by
  \begin{equation*}\label{eqnBetaFunction}
    \beta(m) = \left\{ \begin{array}{ll}
   \bigl( ( \tfrac{m-1}{4} )! \bigr)^2, & \mbox{ if } m\equiv 1\pmod{4}, \\[\medskipamount]
 ( \tfrac{m-3}{4} )! ( \tfrac{m+1}{4} )!, & \mbox{ if } m\equiv 3\pmod{4}.
\end{array} \right.
  \end{equation*}
\end{notation}

The following theorem gives us a lower bound on the number of distinct
$C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn +1$ (or $2mn +3$) for distinct odd integers $m$ and $n$.

\begin{theorem}   \label{thmNumberOfBicentrallyBalancedLabelingsOnRectangle}
  Let $m \geqslant 3$ and $n \geqslant 3$ be distinct odd integers.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+1$ (or $2mn+3$)
  (up to symmetries on the projective plane) is at least
  \begin{equation*}
     \bigl( \tau(m,n) + \tau(n,m) \bigr) 2^{m/2+n/2-3} \beta(m)  \beta(n),
  \end{equation*}
  where $\tau(m,n)$ is the number of distinct $(m,n)$-projective factorization sequences. See Definition \ref{defnProjectiveFactorizationSequence}.
\end{theorem}

\begin{proof}
  Let $X$ be a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
  Let $\eta$ be a permutation on $\{1,2,\ldots,m_0\}$ such that $\eta(i)\equiv i \pmod{2}$ for all $1 \leqslant i \leqslant m_0$
  and $\mathcal{E}_{\eta}(X)$ be the labeling given in Definition \ref{defnElementaryProjectivePermutationColumns}.
  By Lemma \ref{lemmaElementaryProjectivePermutationColumns}, there are $\beta(m)$ distinct bicentrally balanced $C_4$-face-magic projective labelings
  of type $\mathcal{E}_{\eta}(X)$ on $\mathcal{P}_{m,n}$.
  Let $\alpha:\{1,2,\ldots,m_0\}\rightarrow \{0,1\}$ and $\mathcal{E}_{\alpha}(X)$ be the labeling given in Definition \ref{defnElementaryProjectiveSwapColumns}.
  By Lemma \ref{lemmaElementaryProjectiveSwapColumns}, there are $2^{m_0}$ distinct bicentrally balanced $C_4$-face-magic projective labelings
  of type $\mathcal{E}_{\alpha}(X)$ on $\mathcal{P}_{m,n}$.
  Similarly, by Lemma \ref{lemmaElementaryProjectivePermutationRows}, there are $\beta(n)$
  distinct bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ associated with the elementary projective labeling operation given
  in Definition \ref{defnElementaryProjectivePermutationRows}.
  Also, by Lemma \ref{lemmaElementaryProjectiveSwapRows}, there are $2^{n_0}$
  distinct bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ associated with the elementary projective labeling operation given
  in Definition \ref{defnElementaryProjectiveSwapRows}.
  Thus, there are $\beta(m) 2^{m_0} \beta(n) 2^{n_0}$ distinct bicentrally balanced $C_4$-face-magic projective labelings
  on $\mathcal{P}_{m,n}$ that are projective labeling equivalent to $X$.
  For each of these labelings, there are four labelings that are projective labeling equivalent by one of the symmetries
  $R_0$, $R_{180}$, $H$, or $V$ of the projective plane.
  Hence, there are $\tfrac{1}{4} \beta(m) 2^{m_0} \beta(n) 2^{n_0} = 2^{m/2+n/2-3} \beta(m) \beta(n)$ distinct bicentrally balanced $C_4$-face-magic projective labelings
  on $\mathcal{P}_{m,n}$ that are projective labeling equivalent to $X$ up to symmetries of the projective plane.
  By Remark \ref{remBicentrallyBalancedHasValue2MNPlus3} and Lemma \ref{lemmaProjectiveGridSemiC4FaceValueBalance}, a $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$ is bicentrally balanced if and only if its $C_4$-face-magic value is $2mn+3$.
  Therefore, there are $2^{m/2+n/2-3} \beta(m) \beta(n)$ distinct $C_4$-face-magic projective labelings
  on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $2mn+3$ that are projective labeling equivalent to $X$ up to symmetries of the projective plane.

  By Theorem \ref{thmStandardProjectiveC4MagicBicentralConstruction}, each $(m,n)$-projective factorization sequence
  $F$ and each $(n,m)$-projective factorization sequence $F'$ are associated with unique standard
  bicentrally balanced $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$
  given by $\mathrm{HBBL}(F)$ and $\mathrm{VBBL}(F')$, respectively.
  Therefore, there are at least
  \begin{equation*}
    \bigl( \tau(m,n) + \tau(n,m) \bigr) 2^{m/2 +n/2-3}  \beta(m) \allowbreak \beta(n)
  \end{equation*}
  distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ (up to symmetries on the projective plane) that have $C_4$-face-magic value $2mn+3$.

  By Remark \ref{remOrderComplementLabelingCorrespondence}, there are at least
  \begin{equation*}
    \bigl( \tau(m,n) + \tau(n,m) \bigr) 2^{m/2 +n/2-3}  \beta(m) \allowbreak \beta(n)
  \end{equation*}
  distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ (up to symmetries on the projective plane) that have $C_4$-face-magic value $2mn+1$.
\end{proof}

The next proposition gives us a lower bound on the number of distinct
$C_4$-face-magic projective labelings on $\mathcal{P}_{m,m}$ having $C_4$-face-magic value $2m^2 +1$ (or $2m^2 +3$).

\begin{theorem}   \label{thmNumberOfBicentrallyBalancedLabelingsOnSquare}
  Let $m \geqslant 3$ be an odd integer.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,m}$ having $C_4$-face-magic value $2m^2 +1$ (or $2m^2 +3$)
  (up to symmetries on the projective plane) is at least
  \begin{equation*}
     \tau(m,m) 2^{m-3} \bigl( \beta(m) \bigr)^2.
  \end{equation*}
\end{theorem}

The proof of Theorem \ref{thmNumberOfBicentrallyBalancedLabelingsOnSquare} is similar to that of
Theorem \ref{thmNumberOfBicentrallyBalancedLabelingsOnRectangle}.

\begin{theorem} \label{thmNumberOfC4FaceMagicLabelingsOnRectangle}
  Let $m \geqslant 3$ and $n \geqslant 3$ be distinct odd integers.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$
  (up to symmetries on the projective plane) is at least
  \begin{equation*}
    \bigl(  \tau(m,n) + \tau(n,m) \bigr)   2^{m/2 +n/2-3} \bigl( (\tfrac{m-1}{2})! (\tfrac{n-1}{2})!   + 2 \beta(m)  \beta(n) \bigr).
  \end{equation*}
\end{theorem}

\begin{proof}
   By Lemma \ref{lemmaOddByOddProjectiveGridC4FaceValue}, the $C_4$-face-magic value of a $C_4$-face-magic projective labeling on
   $\mathcal{P}_{m,n}$ is either $2mn+1$, $2mn+2$, or $2mn+3$.
   By Theorem \ref{thmNumberOfCentrallyBalancedC4FaceMagicLabelingsOnRectangle}, there are exactly
   \begin{equation*}
    ( \tau(m,n) + \tau(n,m) )2^{m/2 +n/2-3} (\tfrac{m-1}{2})! (\tfrac{n-1}{2})!
   \end{equation*}
   distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$
  (up to symmetries on the projective plane) with $C_4$-face-magic value $2mn+2$.
  By Theorem \ref{thmNumberOfBicentrallyBalancedLabelingsOnRectangle}, there are at least
   \begin{equation*}
    ( \tau(m,n) + \tau(n,m) ) 2^{m/2+n/2-3} \beta(m)  \beta(n)
   \end{equation*}
   distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$
  (up to symmetries on the projective plane) with $C_4$-face-magic value $2mn+1$ (or $2mn+3$).
  The result follows.
\end{proof}

\begin{theorem} \label{thmNumberOfC4FaceMagicLabelingsOnSquare}
  Let $m \geqslant 3$ be an odd integer.
  Then the number of distinct $C_4$-face-magic projective labelings on $\mathcal{P}_{m,m}$
  (up to symmetries on the projective plane) is at least
  \begin{equation*}
    \tau(m,m)  2^{m-3} \bigl(  \bigl( (\tfrac{m-1}{2})! \bigr)^2 + 2 \bigl( \beta(m) \bigr)^2 \bigr).
  \end{equation*}
\end{theorem}

  The proof Theorem \ref{thmNumberOfC4FaceMagicLabelingsOnSquare} is similar to that of Theorem \ref{thmNumberOfC4FaceMagicLabelingsOnRectangle}.

\section{Open Problems}

We conjecture that the labelings given in Theorem \ref{thmStandardProjectiveC4MagicBicentralConstruction} are the only standard bicentrally balanced
$C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.

\begin{conjecture}
Let $m \geqslant 3$ and $n \geqslant 3$ be odd integers.
Suppose $X$ is a standard bicentrally balanced $C_4$-face-magic projective labeling on $\mathcal{P}_{m,n}$.
Then either
\begin{enumerate}
  \item there exists an $(m,n)$-projective factorization sequence $F$ such that $X=\mbox{HBBL}(F)$ or
  \item there exists an $(n,m)$-projective factorization sequence $F'$ such that $X=\mbox{VBBL}(F')$.
\end{enumerate}
\end{conjecture}

Suppose $m\geqslant 3$ and $n \geqslant 3$ are odd integers.
The characterization of $C_4$-face-magic projective labelings  on $\mathcal{P}_{m,n}$ with $C_4$-face-magic value $2mn+2$
is given in \cite{Curran2}.
It is natural to ask if there is a characterization of $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$ when $m\geqslant 2$ and
$n\geqslant 2$ are even integers.
By Lemma \ref{lemmaEvenByEvenProjectiveGridC4FaceValue}, the $C_4$-face-magic value of such a labeling is $2mn+2$.

\begin{problem}
  Let $m\geqslant 2$ and $n\geqslant 2$ be even integers.
  Find a characterization of the $C_4$-face-magic projective labelings on $\mathcal{P}_{m,n}$.
\end{problem}

Curran and Locke \cite{CurranLocke} have characterized the $C_4$-face-magic projective labelings on the $4 \times 4$ projective grid graph $\mathcal{P}_{4,4}$.
They show that there are 144 $C_4$-face-magic projective labelings on $\mathcal{P}_{4,4}$ up to symmetries on the projective plane.

redpajama_set_name: RedPajamaArXiv


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text:
<?xml version="1.0" encoding="UTF-8"?>
<phpunit backupGlobals="false"
         backupStaticAttributes="false"
         bootstrap="vendor/autoload.php"
         colors="true"
         convertErrorsToExceptions="true"
         convertNoticesToExceptions="true"
         convertWarningsToExceptions="true"
         processIsolation="false"
         stopOnFailure="true"
         syntaxCheck="false">
    <testsuites>
        <testsuite name="laraport/blade">
            <directory suffix=".php">./test/</directory>
        </testsuite>
    </testsuites>
</phpunit>

redpajama_set_name: RedPajamaGithub


text:
Q: Running Python file from command line does not load modules I have a Python script that contains the following modules:
from tkinter import *
from tkinter import ttk
from tkinter import filedialog

When I run the code in IDLE by pressing F5 the script runs fine and starts my app.
However, when I go to the command prompt and type
python ScannerApp.py

I get the following error:
File "tkinterTest.py", line 1, in <module>
from tkinter import *
ImportError: No module named tkinter

How do I get rid of this error? The ultimate goal being to make this script into a .exe.
One thought is that python is not added to my environmental variables under Path, it is added as it's own variable. Could that be causing the issue?
My question does not pertain to the difference between Tkinter and tkinter. My question was about why when I ran code through the command line I was getting an error. The issue happened to be that my environmental variable python was set to run python 2.7 instead of the necessary python 3.6 (which uses tkinter).

A: Try adding this for cross compatibility instead of  your previous import code.: (Hoping that's the problem)
try:
    from tkinter import *
    from tkinter import ttk,filedialog
except:
    from Tkinter import *
    from Tkinter import ttk,filedialog

A: Your issue might be that python3 doesn't use Tkinter (with a capital T) but tkinter. That is if you're using pyhton3 of course ^^
https://stackoverflow.com/a/17843652/9368855

A: The solution to my problem was to change the environmental variable python to run version 3.6 instead of 2.7.
The issue was a cross compatibility issue and I found it easier to change the variable instead of trying to have it try both Tkinter and tkinter modules depending on the particular version.

redpajama_set_name: RedPajamaStackExchange


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text:
<?php
	if(isset($_SESSION['Sesion']) && $_SESSION['Sesion']['id_usuario']!="")
	{
		if(isset($_GET['id']))
		{
			$id=$_GET['id'];
			require_once("libs/libs.php");
			$consultar=new Consultar();
			$result=$consultar->_ConsultarInformacionClientePorId($id);
			$fila=$result->fetch_assoc();
		}
?>

        <button type="button" class="close" data-dismiss="modal" aria-label="Close">×</button>

Tipo De Cuerpo

Mesomorfo

                            <input type="radio" id="id_cuerpo" value="MESOMORFO" name="id_cuerpo" <?php if($fila['nb_cuerpo']=="MESOMORFO") echo "checked";?>>

Endomorfo

                            <input type="radio" id="id_cuerpo" value="ENDOMORFO" name="id_cuerpo" <?php if($fila['nb_cuerpo']=="ENDOMORFO") echo "checked";?>>

Ectomorfo

                            <input type="radio" id="id_cuerpo" value="ECTOMORFO" name="id_cuerpo" <?php if($fila['nb_cuerpo']=="ECTOMORFO") echo "checked";?>>

        <button type="button" class="btn btn-default" data-dismiss="modal">Cerrar</button>

                        Usuarios
                        Editar Usuarios

 Inicio

Usuarios

  <button type="button" class="close" data-dismiss="alert">×Close</button>
  Bien Hecho! Usuario Registrado.

                        Formulario de Usuario

                        	En este apartado podrá Editar los usuarios de spin gym, ya sean administradores o entrenadores.

  <button type="button" class="close" data-dismiss="alert">×Close</button>
  Bien Hecho! Cliente Editado.

                    <form name="clientesRegistrar" id="clientesRegistrar" method="post" role="form">

Datos Generales

                                    <input type="hidden" id="id_cliente" value="<?php echo $fila['id_cliente']?>">
                                        Nombres
                                        <input type="text" name="nb_cliente" id="nb_cliente" campo="Nombres" class="form-control requerido"
                                        placeholder="Nombres" value="<?php echo $fila['nb_cliente']?>">

                                        Apellidos
                                        <input type="text" name="nb_apellidos" id="nb_apellidos" campo="Apellido de Usuario" class="form-control requerido"
                                        placeholder="Apellidos" value="<?php echo $fila['nb_apellidos']?>">

                                        Genero del Usuario
                                        <select class="form-control" id="de_genero">
                                            <option value="MASCULINO" <?php if($fila['de_genero']=='MASCULINO') echo "selected"?>>MASCULINO</option>
                                            <option value="FEMENINO" <?php if($fila['de_genero']=='FEMENINO') echo "selected"?>>FEMENINO</option>
                                        </select>

                                        Edad del Usuario
                                          Día
                                        <select id="dia_nacimiento" name="dia_nacimiento"  class="form-control requerido">
                                        <option value="0">Seleccionar...</option>
                                            <?php
                                                for($i=1; $i<=31; $i++)
												{
													if($i==$fila['birth_day'])
													{
														echo '<option value='.$i.' selected>'.$i.'</option>';
													}
													else {echo '<option value='.$i.'>'.$i.'</option>';}
												}
                                            ?>
                                        </select>

                                         Mes
                                           <select  id="mes_nacimiento" name="mes_nacimiento"  class="form-control requerido">
                                                 <option value="0">Seleccionar...</option>
                                                <?php

                                                    for($i=1; $i<=12; $i++)
                                                    {
                                                        switch($i)
                                                        {
                                                            case 1:
                                                                $mes="Enero";
                                                            break;
                                                            case 2:
                                                                $mes="Febrero";
                                                            break;
                                                            case 3:
                                                                $mes="Marzo";
                                                            break;
                                                            case 4:
                                                                $mes="Abril";
                                                            break;
                                                            case 5:
                                                                $mes="Mayo";
                                                            break;
                                                            case 6:
                                                                $mes="Junio";
                                                            break;
                                                            case 7:
                                                                $mes="Julio";
                                                            break;
                                                            case 8:
                                                                $mes="Agosto";
                                                            break;
                                                            case 9:
                                                                $mes="Septiembre";
                                                            break;
                                                            case 10:
                                                                $mes="Ocubre";
                                                            break;
                                                            case 11:
                                                                $mes="Noviembre";
                                                            break;
                                                            case 12:
                                                                $mes="Diciembre";
                                                            break;
                                                        }
														if($i==$fila['birth_month'])
                                                        {echo '<option value='.$i.' selected="selected">'.$mes.'</option>';}
														else {echo '<option value='.$i.' >'.$mes.'</option>';}
                                                    }
                                                ?>
                                          </select>

                                        Año
                                        	<select id="year" name="year" class="form-control requerido">
                                             <option value="0">Seleccionar...</option>
                                                <?php
                                                    for($i=1940; $i<=2008; $i++)
													{
														if($i==$fila['birth_year'])
														{echo '<option value='.$i.' selected="selected">'.$i.'</option>';}
														else {echo '<option value='.$i.'>'.$i.'</option>';}
													}
                                                ?>
                                            </select>

                                        Email Usuario
                                        <input type="email" name="de_email" id="de_email" campo="Email de Cliente" class="form-control requerido"
                                        placeholder="Email del Usuario" value="<?php echo $fila['de_email']?>">

                                        Telefono Usuario:

                                            <input type="text" name="num_telefono" id="num_telefono" class="form-control" data-inputmask='"mask": "(999) 999-9999"' data-mask
                                            value="<?php echo $fila['num_telefono']?>">

                                        Celular Usuario:

                                            <input type="text" name="num_celular" id="num_celular" campo="Celular de Cliente" class="form-control requerido"
                                            data-inputmask='"mask": "(999) 999-9999"' data-mask value="<?php echo $fila['num_celular']?>">

                                     Los campos marcados con * son obligatorios.

Datos de Ubicación

                                        Colonia
                                        <input type="text" name="de_colonia" id="de_colonia" class="form-control" placeholder="Colonia del Usuario" value="<?php echo $fila['de_colonia']?>">

                                        Dirección
                                        <textarea name="de_domicilio" id="de_domicilio" campo="Dirección de Cliente" class="form-control requerido" rows="3"
                                        placeholder="Dirección del Usuario"> <?php echo $fila['de_domicilio']?></textarea>

                                        Códico Postal
                                        <input type="text" name="num_codigoPostal" id="num_codigoPostal" class="form-control"
                                         placeholder="Código Postal del Usuario" value="<?php echo $fila['num_codigoPostal']?>">

                                        Tipo De Cuerpo
                                        <input type="button" class="btn btn-info" onclick="TipoCuerpo()" value="Seleccionar" >

                                    Los campos marcados con * son obligatorios.

<button type="button" class="btn btn-primary" onclick="checkRequeridosEditar();">EDITAR USUARIO</button>

<button type="reset" class="btn btn-danger" onclick="window.location='index.php?nav=Clientes';">CANCELAR</button>

                    </form>

<?php
	include('includes/footer.php');
	include("modulos/Usuarios/JavaScript.php");
?>
<?php } else {header("Location: Login.php");}?>


John Mink Resume | Architectural Partnership, Inc.
Thirty-five years of experience in architecture and construction, with an emphasis on historic preservation, custom homes, residential remodeling, entertainment and retail design.
American Institute of Architects / Ongoing Continuing Education / Graduate School of Design, Harvard University, Cambridge, Massachusetts.
Master of Environmental Planning / College of Architecture, Arizona State University, Tempe, Arizona. l982.
Bachelor of Environmental Design / College of Environmental Design, University of Colorado, Boulder, Colorado. l978.
Architectural Partnership, Inc., 4843 Country Club Way, Boulder, CO. Architect / President. 1992- present.
Knudson Gloss Architects, 3050 Broadway, Suite 301, Boulder, CO. Associate. l988 - 1992.
Milburn, Sparn, McEvoy, Railton & Coburn, Inc., Architecture and Planning, 3300 Mitchell Lane, Suite 390, Boulder, CO. Associate. l987-1988.
Knudson Associates Architects, 3050 Broadway, Suite 301, Boulder, CO. Job Captain. l983-l987.
United States Department of Energy, College of Architecture, Arizona State University. Research Associate. Tempe, AZ . l981-l982.
Self employed framing and trim carpenter, Boulder, CO. 1974-1981.


This track will invite papers that address and reflect upon community grounded ICT developments in theory and practice in an effort to strengthen current efforts redefining a collaborative research and development paradigm with indigenous and local communities and promoting sustainable information society.
Although many ICT developments target local communities' needs however, they are often being determined by externals stakeholders insufficiently familiar with the local context. This has led to numerous "white elephant" projects and millions wasted. Thus it has been established that rather than developing "for the community", ICT solutions should be developed "with the community" to enhance social harmony, community gains and impact. This process however is not straightforward and brings about numerous challenges in terms of collaboration, timing, processes, expectations, skills, agendas, conceptual and technical understandings and epistemologies.
This track will therefore focus on community grounded projects where partnerships between communities and other stakeholders have been established to situate the ICT developments within the communities' framework. In this context it is paramount to reconsider current project success and evaluation criteria, methods and tools and their situated appropriateness as well as their contextual transferability, and most importantly (re)defining cultural and research protocols to create an enabling environment.
Evaluation: The papers should address related questions such as, what makes a collaborative community grounded ICT development project successful? How is collective success defined by the different stakeholders? In which way should community and academic learning and transformation through engagement in longitudinal research projects be holistically evaluated besides technological criteria? Thus we encourage submitting studies drawing on models and evaluation criteria for community grounded ICT development projects including socio-economic and environmental impacts.
Methods and tools: Papers should present methods and tools that are appropriate and novel shaped within a collaborative context. Welcome are also reflective reports on success and failures of deployed methods which can expand current attempts to establish community-based development method repository. Addressing questions of transferability of methods and tools versus situated validity is of particular interest towards theory building.
Engagement Protocols: Community grounded research and development requires a major paradigm shift towards establishing a coherent development context. Thus cultural, research and development protocols need to be (re)defined and negotiated among the stakeholders addressing issues such as commitments, responsibilities, and benefits among others. Within the engagement landscape we are also looking at the preparedness of the parties involved, mutual learning and transformations within the process.


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For more on the sale, go to Good Morning America online for the full report.
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