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\section{Introduction}

The $K$-nucleon ($KN$) interactions in nuclear matter have been an
interesting topic in nuclear physics, since Kaplan firstly discussed
the question of kaon condensation in dense nuclear matter about
twenty years ago \cite{Kaplan}. With several typical methods, such
as chiral perturbation theory (ChPT) \cite{c5,brown,c4}, RMF model
\cite{c1,c4,c5,c3}, chiral coupled channel model \cite{Weis,d4},
chiral unitary model \cite{oset,Hirenzaki} or other chiral model
\cite{a2,a3}, and the phenomenological model by fitting the $K^{-}$
atomic data using a density-dependent optical potential (DD model)
\cite{Frid,Frid1999}, a strong attractive $K^{-}$-nucleus potential
at threshold is predicted. The $K^{-}$-nucleus potential strongly
depends on the model used. The DD model gives the deepest inner
$K^{-}$-nucleus potential which is in the range of $150\sim200$ MeV
\cite{Frid,Frid1999}. The chiral coupled channel model predicts the
$K^{-}$-nucleus potential in the range of $85\sim 120$ MeV
\cite{Weis,c3,c4}, close to that by RMF model (The $\sigma$-$K$
coupling constant determined by the $KN$ scatering length). The
chiral models give shallower $K^{-}$-nucleus potential $50\sim 70$
MeV \cite{c4,oset,Hirenzaki,a2,a3}.

Based on the strong attractive $K^{-}$-nucleus potential, there
maybe exist ``deeply bound kaonic atoms'' or kaonic nuclei. The
existence of the ``deeply bound kaonic atoms'', or, kaonic nuclei
was already well-known in later 1990s \cite{deepbound, a4}. It was
suggested that the kaonic nuclei could be produced by the ($K^-$,
$p$) and ($K^-$, $n$) reactions \cite{a4}. Since then, the issue on
the $K^-$-nucleus deeply bound states (kaonic nuclei) has been a hot
topic.

Theoretically, there have been many works on kaonic nuclei. The
binding energies, the widths, the production methods, the nuclear
densities around $K^-$-meson, and other aspects were discussed with
different models
\cite{a2,a6,a3,a8,a9,a10,a11,d,Galaa,Agnello,Dote,Yamagata,OsetToki}.
Experimentally, some evidences for the candidate states in the
($K_{\mathrm{stop}},\ n)$ and ($K_{\mathrm{stop}},\ p)$ reactions on
$^4$He \cite{b1,b2}, and in the ($K^-$, $n$) in-flight reaction on
$^{16}$O \cite{b3} are reported. Recently, by using the FINUDA
spectrometer installed at the e$^+$e$^-$ collider DA$\Phi$NE, the
FINUDA collaboration succeeded in detecting a $K^-$ bound state
$K^-pp$ through its two-body decay into a $\Lambda$\ hyperon and a
proton \cite{b4}. However, just some months ago, Magas \emph{et al.}
reanalyzed the FINUDA data and claimed that the experimental
spectrum can be naturally explained in their Monte Carlo simulation
of the $K^-$ absorption events in nuclei, without the need of exotic
mechanisms like the formation of a $K^-pp$ bound state \cite{b5}.
Thus, further experimental and theoretical works are needed on this
issue.

In the present work, we will study the static properties of the
possible kaonic nuclei in the framework of relativistic mean-field
theory (RMF). The pioneer works, describing the in-medium $K^-N $
interaction with RMF method, can be found in  \cite{c1,c2,c3}. Since
then, the RMF model had been widely used to describe the in-medium
properties of (anti)kaon mesons \cite{c4,c5}, the $K^-$-nucleus
interaction
 \cite{Frid1999}, the $K^-$-nucleus elastic scattering \cite{c7} and
the kaonic nuclei \cite{d,Galaa}. In this work, the static
properties of kaonic nuclei are obtained by solving,
self-consistently, the equations of motion for nucleons and mesons,
which are derived from the Lagrangian for nucleons and (anti)kaon
mesons. To describe the widths of the $K^-$ in nuclei, the equation
of motion for $K^-$ are modified by introducing an imaginary part of
the self-energy phenomenologically. In the calculations, we will
focus on the energy spectra, the nuclear density distribution, the
r.m.s.\ radii of proton, neutron and charge distribution for the
possible kaonic nuclei.

The paper is organized as follows. In the subsequent section, the
Lagrangian density is given, the equations of motion for nucleons
and the meson fields $\sigma$, $\omega$, $\rho$, and Kaons are
deduced, the imaginary part of the self-energies are introduced and
the binding energies of $K^-$ are defined. We then present our
results and discussions of the obtained properties of kaonic nuclei
in Sec. III. Finally a summary is given in Sec. IV.

\section{Formulas}

In relativistic mean field theory, the standard Lagrangian density
for an ordinary nucleus can be written as \cite{d1,d2}
\begin{eqnarray}
{\mathcal{L}}_{0}
={\mathcal{L}}_{\mathrm{Dirac}}
 +{\mathcal{L}}_{\sigma}
 +{\mathcal{L}}_{\omega}
 +{\mathcal{L}}_{\rho}
 +{\mathcal{L}}_{A},
\end{eqnarray}
where
\begin{eqnarray}
{\mathcal{L}}_{\mathrm{Dirac}}
&=&
 \bar{\Psi}_N(i\gamma^{\mu}\partial_{\mu}-M_{N})\Psi_N, \\
{\mathcal{L}}_{\sigma}
&=&
 \frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma
 -\frac{1}{2}m_{\sigma}^2\sigma^2
 -g_{\sigma N }\bar{\Psi}_N\sigma\Psi_N
\nonumber\\
& &
 -\frac{1}{3}g_{2}\sigma^3
 -\frac{1}{4}g_{3}\sigma^4,\\
{\mathcal{L}}_{\omega}
&=&
 -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
 +\frac{1}{2}m_{\omega}^2\omega_{\mu}\omega^{\mu}
\nonumber\\
& &
 -g_{\omega N}\bar{\Psi}_N\gamma^{\mu}\Psi_N\omega_{\mu}, \\
{\mathcal{L}}_{\rho}
&=&
 -\frac{1}{4}\vec{G}_{\mu\nu}\vec{G}^{\mu\nu}
 +\frac{1}{2}m_{\rho}^2
\vec{\rho}_{\mu}\cdot\vec{\rho}^{\mu}\nonumber\\
& & -g_{\rho N}\bar{\Psi}_{N}\vec{\rho}^{\mu}\cdot \vec{I}\Psi_{N},\\
{\mathcal{L}}_{A}
&=&
  -\frac{1}{4}H_{\mu\nu}H^{\mu\nu}
  -e\bar{\Psi}_N\gamma_{\mu}I_{c}A^{\mu}\Psi_N,
\end{eqnarray}
with
\begin{eqnarray}
F_{\mu\nu}
&=&
 \partial_{\nu}\omega_{\mu}-\partial_{\mu}\omega_{\nu},\\
\vec{G}_{\mu\nu} &=&
 \partial_{\nu}\vec{\rho}_{\mu}-\partial_{\mu}\vec{\rho}_{\nu}, \\
H_{\mu\nu}
&=& \partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu},
\end{eqnarray}
where the meson fields are denoted by $\sigma$, $\omega_{\mu}$,
and $\vec{\rho}_{\mu}$, and their masses by $m_{\sigma}$,
$m_{\omega}$, $m_{\rho}$, respectively.  $\Psi_N$ is the nucleon
field with corresponding mass $M_N$. $A_{\mu}$ is the
electromagnetic field. $g_{\sigma N} $, $g_{\omega N} $, and
$g_{\rho N} $ are, respectively, the $\sigma$-$N$, $\omega$-$N$,
and $\rho$-$N$ coupling constants. $I_{c}=(1+\tau_{3})/2$ is the
Coulomb interaction operator with $\tau_3$ being the third
component of the isospin Pauli matrices for nucleons. $I$ is the
nucleon isospin operator. In this paper, we adopt the NL-SH
parameter set \cite{d3}, which describes the properties of finite
nuclei reasonably. The masses and coupling constants are listed in
Tab. \ref{paramtab}.

For a $K^-$-nucleus system, another Lagrangian density
${\mathcal{L}}_{\mathrm{K}}$ describing the (anti)kaon interaction
with nucleons should be added to ${\mathcal{L}}_0$. Kaons are
incorporated into the RMF model by using the $KN$ interactions
motivated by the one-meson-exchange models \cite{d0}. In the
meson-exchange picture, the scalar and vector interactions between
kaons and nucleons are mediated by the exchange of $\sigma$ and
$\omega$ mesons, respectively.  The coupling of the (anti)kaon to
the isovector $\rho$ meson is here excluded due to $N= Z$ nuclear
cores. Thus, the simplest kaon-meson interaction Lagrangian density
${\mathcal{L}}_{\mathrm{K}}$ is written as \cite{c3}
\begin{eqnarray}
{\mathcal{L}}_{\mathrm{K}}
&=&
 \partial_{\mu}\bar{K}\partial^{\mu}K
 -m_K^2\bar{K}K-g_{\sigma K}m_{K}\bar{K}K\sigma
\nonumber\\
& &
 -ig_{\omega K}(\bar{K}\partial_{\mu}K-K\partial_{\mu}\bar{K})\omega^{\mu}
\nonumber\\
& &
 +(g_{\omega K}\omega^{\mu})^2\bar{K}K,
\end{eqnarray}
where $g_{\sigma\mathrm{K}}$ and $g_{\omega\mathrm{K}}$ are the
$\sigma$-$K$\ and $\omega$-$K$ coupling constants.
$g_{\omega\mathrm{K}}$ is chosen from the SU(3) relation assuming
ideal mixing, i.e., $2g_{\omega\mathrm{K}}=2g_{\pi\pi\rho}=6.04$.
$g_{\sigma\mathrm{K}}$ can be obtained from several methods, e.g.
the Bonn model \cite{d0}, by reproducing the strongly attractive
potential seen in kaonic atoms \cite{Frid1999} or by fitting the
$KN$ scattering lengths in experiments \cite{c3,c4}. In this paper,
we take the modest $K$-$\sigma$ coupling, $g_{\sigma K}  =2.088
(\approx g_{\sigma N}/5)$, by fitting the experimental $KN$
scattering length \cite{c4}. The advantages of this method were
pointed out in \cite{c4}. With these determined coupling constants,
the antikaon optical potential at normal nuclear density is about
$U_{\mathrm{K}^-}=-(85\sim 100)$ MeV \cite{c3}, which is compatible
with several groups' predictions. For example, Akaishi \emph{et al.}
gave $-119$ MeV for a $K^-$ in nuclear matter at the normal density
\cite{a9}, and Weise's gave $U_{\mathrm{K}^-}=-(120\sim 130)$ MeV by
using chiral $KN$ interaction \cite{d4}. It is interesting that the
recent experiment predicts the in-medium $K^-N$ potential is on the
order of $-80$ MeV at normal nuclear density \cite{Sch2006}, which
is also agreement with the $K^-N$ potential adopted in present
paper.

\begin{table}[ht]
\begin{center}
\caption{Parameters used in the present calculations.}
\label{paramtab}
\begin{tabular}{|cccc|cccc|}\hline\hline
\multicolumn{4}{|c|}{masses (MeV)}
     & \multicolumn{4}{|c|}{couplings}  \\ \cline{2-3}\cline{6-7}
$M_N$ & $m_{\sigma}$ & $m_{\omega}$ & $m_{\rho}$
                 & $g_{\sigma N} $ & $g_{\omega N} $
                 & $g_{\rho N} $   & $g_3$ \\ \hline
939.0 & 526.059 & 783.0 & 763.0 & 10.444 & 12.945 & 8.766 & -15.8337 \\
\multicolumn{4}{|c|}{\mbox{}}
    & \multicolumn{4}{|c|}{$g_2=-6.9099$ fm$^{-1}$}  \\ \hline\hline
\end{tabular}
\end{center}
\end{table}

In the mean field approximation, the meson-fields $\sigma$,
$\omega_{\mu}$, and $\rho_{\mu}$, and the photons $A_{\mu}$ are
replaced with their mean values, $\langle \sigma \rangle$, $\langle
\omega_{\mu}\rangle$, $\langle\rho_{\mu}\rangle$ and $\langle
A_{\mu}\rangle$, respectively. For a spherical nucleus, only the
mean values of the time components $\langle \omega_{0}\rangle$,
$\langle\rho_{0}\rangle$ and $\langle A_{0}\rangle$ remain, which
are denoted by $\omega_{0}$, and $\rho_{0}$, and $A_{0}$
respectively. Then the equations of motion for nucleons, $\omega$,
$\sigma$, $\rho$, and photons are

\begin{widetext}
\begin{eqnarray}
& \left[-i\vec{\alpha}\cdot \vec{\nabla}+\beta(M_{N}+g_{\sigma
N}\sigma_{0})+g_{\omega N}\omega_{0} +g_{\rho N}\tau_3\rho_{0}+e
I_{c}A_{0}\right]\Psi_{N}=\epsilon \Psi_{N},
&\label{eossimp1}\\
&\left(-\nabla^2+m_{\sigma}^2\right)\sigma_{0}= -g_{\sigma N}
\bar{\Psi}_{N}\Psi_{N}
-g_{2}\sigma_{0}^2-g_{3}\sigma_{0}^3-g_{\sigma K}m_{K} \bar{K}K ,
&\label{eossimp2}\\
& \left(-\nabla^2+m_{\omega}^2\right)\omega_{0}=g_{\omega N}
\bar{\Psi}_{N}\gamma^{0} \Psi_{N} -2g_{\omega K} (E+g_{\omega
K}\omega_{0})\bar{K}K ,
&  \label{eossimp3}\\
& \left(-\nabla^2+m_{\rho}^2\right)\rho_{0}=g_{\rho N}
\bar{\Psi}_{N}\gamma^{0}I \Psi_{N} ,
& \label{eossimp4}\\
& -\nabla^2A_0=e \bar{\Psi}_{N}\gamma^{0}I_{c} \Psi_{N}, &
\label{eossimp5}
\end{eqnarray}
\end{widetext}
and the equation of motion for antikaon is
\begin{eqnarray}
\left[-\nabla^2+(m_{\mathrm{K}}^2-E^2)+\Pi \right] \bar{K} =0,
\label{eosfin5}
\end{eqnarray}
with the antikaon self-energy in nuclei
\begin{eqnarray}
\Pi =-2g_{\omega\mathrm{K}}E\omega_0
+g_{\sigma\mathrm{K}}m_{\mathrm{K}}\sigma_{0}
-(g_{\omega\mathrm{K}}\omega_0)^2. \label{self}
\end{eqnarray}
In the above equations, $\epsilon$ is the nucleon single-particle
energy, and $E$ is the single-particle energy for antikaon meson. We
can see that the antikaon meson does not relate the Dirac equation
for nucleons directly. However, the presence of antikaon leads to
additional source terms in the equations of motion for meson fields
of $\sigma$ and $\omega$. Thus, the antikaon meson affects the
nucleon fields by changing the strength of meson fields of $\sigma$
and $\omega$ indirectly.

So far, we ignored the antikaon meson absorption in the nucleus,
which requires a complex potential. Within the framework of RMF
model, we are not able to calculate the imaginary part of the
potential directly. In order to include the effects of the antikaon
meson absorption in the nucleus on the calculations, and make a more
realistic estimate for the calculated results, we assume a specific
form for the antikaon self-energy with an imaginary part
\begin{eqnarray}
\widetilde{\Pi} &=&\left[-2g_{\omega\mathrm{K}}\omega_0\mathrm{Re}E
+g_{\sigma\mathrm{K}}m_{\mathrm{K}}\sigma_{0}
-(g_{\omega\mathrm{K}}\omega_0)^2\right] \nonumber \\
&& +i\left[-2(\mathrm{Re}E)f V_{0} \frac{\rho}{\rho_0}\right].
\label{selfw}
\end{eqnarray}
The similar method is also used to analyze the widths of mesonic
nuclei in \cite{d,Ludh}.

In this work, the imaginary part of the potential $\mathrm{Im}U$ is
given with the simple ``$t\rho$" form, namely,
$\mathrm{Im}U=-V_{0}\rho/\rho_0$. Where $V_{0}$ is the imaginary
potential depth at normal nuclear density, which strongly depends on
the model adopted.  The largest value $V_{0} \sim 50$ MeV is given
by fitting the experimental data of the kaonic
atoms\cite{Frid,Frid1999}, and the chiral model (only the antikaons
are dressed self-consistently) \cite{01im}. While, by fitting the
experimental data of the antikaon-nucleus scattering, the imaginary
potential depth become much shallower: $V_{0}\sim 35$ MeV \cite{c7}.
Another much shallower imaginary potential depth $20\sim 25$ MeV is
predicted by Ramos \emph{et al.} with the meson-exchange model. The
shallowest imaginary potential depth, $\sim 15$ MeV, is given by the
chiral model which incorporates the dressing of the pion and
antikaons \cite{01im}. Thus, in the present work, we set the
imaginary potential depth $V_{0}$ in the range of $15\sim 50$ MeV.

On the other hand, the phase space available for the decay
products should be reduced for deeply bound states, which will
decrease the imaginary potentials (widths). Thus, a suppression
factor, $f$, multiplying $\mathrm{Im}U$ were suggested to be
introduced by Mare\v{s} \emph{et al.} \cite{d,Galaa}. In our
calculations, we adopt their method as well. In this method, two
decay channels are considered. One is the mesonic decay channel,
$\bar{K} N\rightarrow \pi \Sigma,\ \ \pi \Lambda$. The
corresponding suppression factor is given by
\begin{widetext}
\begin{eqnarray}
f_1=\frac{M_{01}^3}{M_1^3}\sqrt{\frac{[M_1^2-(m_{\pi}+M_Y)^2][M_1^2-(m_{\pi}-M_Y)^2]}
{[M_{01}^2-(m_{\pi}+M_Y)^2][M_{01}^2-(m_{\pi}-M_Y)^2]}}
\Theta(M1-m_{\pi}-M_Y),
\end{eqnarray}
\end{widetext}
where $M_{01}=m_k+M_N$, $M_{1}=\mathrm{Re}E+M_N$ and
$Y=\Sigma,\ \Lambda$. The other channel is the non-mesonic decay
channel, $\bar{K} NN\rightarrow YN$, and the corresponding
suppression factor is
\begin{widetext}
\begin{eqnarray}
f_2=\frac{M_{02}^3}{M_2^3}\sqrt{\frac{[M_2^2-(M_{N}+M_Y)^2][M_2^2-(M_{N}-M_Y)^2]}
{[M_{02}^2-(M_{N}+M_Y)^2][M_{02}^2-(M_{N}-M_Y)^2]}}\Theta
(M_2-M_{N}-M_Y),
\end{eqnarray}
\end{widetext}
where $M_{02}=m_k+2M_N$, $M_{2}=\mathrm{Re}E+2M_N$. Since $\Sigma$
final states dominate both the mesonic  and non-mesonic decay
channels \cite{chann}, in the calculations, the hyperon $Y$ is set
as $Y=\Sigma$. The suppression factor $f$ can be assumed a mixture
of 80\% mesonic decay and 20\% non-mesonic decay \cite{chann}, thus
\begin{eqnarray}
f=0.8 f_1+0.2f_2.
\end{eqnarray}

In the calculations, in order to include the effects from the
``imaginary potential", we use the modified Klein-Gordon equation
\begin{eqnarray}
\left[-\nabla^2+(m_{\mathrm{K}}^2-E^2)+\widetilde{\Pi}  \right]
\bar{K} =0. \label{mdf}
\end{eqnarray}
The complex eigenenergies are
\begin{eqnarray}
E=-B^{s,p}_K+m_K-i\Gamma/2,
\end{eqnarray}
where the real part corresponds to the single-particle $K^-$ binding
energy, which is defined as
\begin{eqnarray}
B^{s,p}_{\mathrm{K}}=m_{\mathrm{K}}-\mathrm{Re}E,
\end{eqnarray}
and the imaginary part of the complex eigenenergies corresponds to
the widths
\begin{eqnarray}
\Gamma=-2\mathrm{Im} E.
\end{eqnarray}

The binding energy of $K^-$, $B_{K^{-}}$, is defined as the
difference between the total binding energy of the $K^-$-nucleus
system, $B(^{A}ZK^{-})$, and the total binding energy of the
ordinary nucleus, $B(^{A}Z)$ \cite{Galaa}:
\begin{eqnarray}
B_{\mathrm{K}}=B(^{A}ZK^{-})-B(^{A}Z).
\end{eqnarray}
In RMF, from the Lagrangian density (Eqs.(1---6)) we can deduce the
energy-momentum tensor $T^{\mu\nu}$ and then obtain the total
binding energy of the ordinary nucleus at once:
\begin{eqnarray}\label{hdd}
B(^{A}Z)&=&A\times
M_{N}+E_{c.m.}-\sum_{i=1}^{A}\epsilon_i\\\nonumber
&&-\frac{1}{2}\int d\textbf{r}\bigg\{-g_{\sigma
N}\sigma_0\bar{\Psi}_{N}\Psi_{N} -g_{2}\sigma_{0}^2 \\ \nonumber
&&-g_{3}\sigma_{0}^3-g_{\omega N}\omega_0
\Psi^{\dagger}_{N}\Psi_{N}\\ \nonumber
&&-g_{\rho N}
\bar{\Psi}_{N}\gamma^{0}I \Psi_{N}-e \bar{\Psi}_{N}\gamma^{0}I_{c}
\Psi_{N}\bigg\},
\end{eqnarray}
where $\epsilon_i$ is the nucleon single-particle energy labelled
$i$, and $E_{c.m.}=3/4 \cdot 41A^{1/3}$ is the center-of-mass
energy. Using the same method, according to the Eqs.(1---6) and
(10) we obtain the total binding energy of the $K^-$-nucleus
system:
\begin{eqnarray}\label{hkdd}
B(^{A}ZK^{-})&=&A\times
M_{N}+E_{c.m.}+B^{s,p}_{\mathrm{K}}\\
\nonumber &&-\sum_{i=1}^{A}\epsilon_i-\frac{1}{2}\int
d\textbf{r}\bigg\{-g_{\sigma N}\sigma_0\bar{\Psi}_{N}\Psi_{N}
\\
\nonumber &&-g_{2}\sigma_{0}^2-g_{3}\sigma_{0}^3 -g_{\omega N}\omega_0\Psi^{\dagger}_{N}\Psi_{N}\\
\nonumber &&
-g_{\rho N} \bar{\Psi}_{N}\gamma^{0}I \Psi_{N}-e \bar{\Psi}_{N}\gamma^{0}I_{c} \Psi_{N} \\
\nonumber &&+2g_{\omega K}\omega_0(3\mathrm{Re} E+g_{\omega
K}\omega_0)\bar{K}K \\
\nonumber &&-g_{\sigma K}m_{K}\sigma_{0}\bar{K}K \bigg\}.
\end{eqnarray}
From the above, we can see that for the appearance of the antikaon
there are additional terms, $-g_{\sigma K}m_{K}\sigma_{0}\bar{K}K$
and $2g_{\omega K}(3\mathrm{Re} E+g_{\omega K}\omega_0)\omega_0$, in
the integral of Eq.(\ref{hkdd}) compared with those of
Eq.(\ref{hdd}).

Solving the equations (\ref{eossimp1})--- (\ref{eossimp5}) and
Eq.(\ref{mdf}) self-consistently, we can obtain the properties of
$K^{-}$-nuclei. One point must be noted that the antikaon energy $E$
in Eq.(\ref{eossimp3}) should be replaced with its real part,
$\mathrm{Re} E$, in the calculations.

\section{Results and discussions}

The main purpose of the present calculations is to obtain the
properties of $K^-$ nuclei in RMF, such as the binding energies, the
nuclear density distributions and the r.m.s.\ radii of proton,
neutron and charge distributions. We carry out calculations for the
$N=Z$ even-even nuclear cores $^{12}$C$K^-$, $^{16}$O$K^-$,
$^{20}$Ne$K^-$, $^{24}$Mg$K^-$, $^{28}$Si$K^-$, $^{32}$S$K^-$,
$^{36}$Ar$K^-$, $^{40}$Ca$K^-$ and $^{44}$Ti$K^-$, respectively. For
the imaginary potential depths at normal nuclear density are not
well determined, in the calculations, we choose three values,
$V_{0}=15$, $30$ and $50$ MeV, respectively. The results are shown
in Tab.\ \ref{Calll}, \ref{Cal} and Figs.\ \ref{Dens}.

\begin{widetext}
\begin{center}
\begin{table}[ht]
 \caption{The single-particle $K^-$ binding energies, $B^{s,p}_{K^-}=m_K-ReE$; the binding energy of $K^-$,
 $B_{K^{-}}$; the total binding energy of the $K^-$-nucleus system,
 $B(^{A}ZK^{-})$; the total binding energy of the ordinary nucleus, $B(^{A}Z)$
 and the widths, $\Gamma$, (all in MeV), in various nuclei, where the
 complex eigenenergies are, $E=-B^{s,p}_K+m_K-i\Gamma/2$.}\label{Calll}
\begin{tabular}{ccccccc|cccc|cccccc}  \hline \hline
 &&&\multicolumn{3}{c}{\underline{$V_{0}=15$ (MeV)}}& &\multicolumn{3}{c}{\underline{$V_{0}=30$ (MeV)}}& &\multicolumn{3}{c}{\underline{$V_{0}=50$ (MeV)}}
 \\
&&$B(^AZ)$& $B(^AZK^-)$ & $B_K$\ \ \ &$B^{s,p}_K$& $\Gamma$ &
$B(^{A}ZK^-)$ & $B_K$\ \ & $B^{s,p}_K$ &$\Gamma$ &$B(^{A}ZK^-)$ &
$B_K$ \ \ & $B^{s,p}_K$ & $\Gamma$
\\ \hline \hline
 \\
$^{12}$C$K^-$    &$1s$& 89.6   &177.6 &\textbf{88.0} \ \ &102.3\ \ &11.2   &177.0&\textbf{87.4}  \ \ &100.4\ \  &19.4   &175.6&\textbf{86.0}    \ \  &97.6 \ \  &44.0&\\
                 &$1p$&        &115.2 &\textbf{25.6} \ \ &35.1 \ \ &23.5   &114.5&\textbf{24.8}  \ \ &33.2 \ \  &47.0    &111.2&\textbf{21.6}   \ \  &28.8 \ \  &80.9&\\
\\
$^{16}$O$K^{-}$  &$1s$& 128.5  &203.8 &\textbf{75.3} \ \ &88.3 \ \ &16.2   &203.4&\textbf{74.8}  \ \ &86.7 \ \   &32.8    &202.4 & \textbf{73.9}\ \  &84.4 \ \  &56.4\\
                 &$1p$&        &158.7 &\textbf{30.2} \ \ &41.1 \ \ &21.1   &157.7&\textbf{29.2}  \ \ &38.5 \ \   &43.0    &156.9 & \textbf{28.4}\ \  &36.6 \ \  &72.8\\
\\
$^{20}$Ne$K^{-}$ &$1s$& 142.6  &221.0 &\textbf{78.4} \ \ &90.4 \ \ &14.8    &220.8& \textbf{78.2}\ \ &89.4 \ \    &30.0    &220.0& \textbf{77.4}\ \  &87.7 \ \  &51.4\\
                 &$1p$&        &180.2 &\textbf{37.7} \ \ &48.9 \ \ &21.5    &179.9& \textbf{37.3}\ \ &47.5 \ \    &42.8    &178.5& \textbf{35.9}\ \  &45.3 \ \  &71.5\\
\\
$^{24}$Mg$K^{-}$ &$1s$&182.8   &267.5 &\textbf{84.7} \ \ &95.3 \ \ &12.2    &267.4& \textbf{84.6}\ \ &94.7 \ \   &25.2    &266.8& \textbf{84.0} \ \  &93.3 \ \  &43.6\\
                 &$1p$&        &231.0 &\textbf{48.2} \ \ &59.4 \ \ &22.0    &230.7& \textbf{47.9}\ \ &58.5 \ \   &42.6    &229.1& \textbf{46.3} \ \  &56.4 \ \  &74.0\\
\\
$^{28}$Si$K^{-}$ &$1s$&231.6   &322.4 &\textbf{90.8} \ \ &100.2\ \ &8.2    &322.2& \textbf{90.6} \ \ &99.6 \ \   &17.8    &321.8& \textbf{90.2} \ \  &98.7 \ \  &32.6\\
                 &$1p$&        &289.0 &\textbf{56.9} \ \ &68.4 \ \ &20.3    &288.2& \textbf{56.6}\ \ &67.2 \ \   &40.7    &287.0& \textbf{55.4} \ \  &65.7 \ \  & 68.6\\
\\
$^{32}$S$K^{-} $ &$1s$&261.6   &358.1 &\textbf{96.5} \ \ &107.8\ \ &14.2   &358.1& \textbf{96.5} \ \ &106.8\ \    &27.0   &357.8& \textbf{96.2} \ \  &104.1\ \  &37.0\\
                 &$1p$&        &320.1 &\textbf{58.5} \ \ &67.5 \ \ &20.2    &319.8& \textbf{58.2}\ \ &66.9 \ \    &40.6    &317.3& \textbf{55.7}\ \  &66.3 \ \  &68.8\\
\\
$^{36}$Ar$K^{-}$ &$1s$&295.2   &387.8 &\textbf{92.6} \ \ &102.4\ \ & 9.2   &387.7& \textbf{92.5} \ \ &101.3\ \   &14.4   &387.3& \textbf{92.1}  \ \  &100.0\ \  &27.8\\
                 &$1p$&        &352.9 &\textbf{57.7} \ \ &68.1 \ \ &19.2    &352.6& \textbf{57.4}\ \  &67.4\ \    &39.1   &351.6& \textbf{56.3} \ \  &66.1 \ \  &66.0\\
\\
$^{40}$Ca$K^{-}$ &$1s$&340.6   &430.4 &\textbf{89.8} \ \ &99.0 \ \ &9.3     &430.3& \textbf{89.7}\ \  &98.0\ \    &20.1    &429.7& \textbf{89.2}\ \   &96.4\ \  &36.3\\
                 &$1p$&        &400.6 &\textbf{60.0} \ \ &69.3 \ \ &18.6    &400.4& \textbf{59.8}\ \  &68.7\ \    &37.3    &399.5& \textbf{59.0}\ \  &67.6 \ \  &62.9\\
\\
$^{44}$Ti$K^{-}$ &$1s$&365.6   &455.8 &\textbf{90.3} \ \ &98.5 \ \ &9.4     &455.5& \textbf{90.0}\ \  &97.7\ \    &19.9    &454.7& \textbf{89.2}\ \  &96.3 \ \   &35.7\\
                 &$1p$&        &428.8 &\textbf{63.6} \ \ &72.3 \ \ &18.0    &428.5& \textbf{63.0}\ \  &71.7\ \    &36.1    &427.6& \textbf{62.1}\ \  &70.7 \ \  &61.0\\
\\
 \hline \hline
\end{tabular}
\end{table}
\end{center}
\end{widetext}

\subsection{Single-particle energies and widths }

The single-particle $K^-$ binding energies, $B^{s,p}_{K^-}$, the
binding energy of $K^-$, $B_{K^{-}}$, the total binding energy of
the kaonic nucleus, $B(^{A}ZK^{-})$, the total binding energy of
the ordinary nucleus, $B(^{A}Z)$ and the widths, $\Gamma$ are
listed in Tab.\ \ref{Calll}. From the Tab. we find that the
imaginary potentials (namely, the width) have a few effects on the
values of $B^{s,p}_{K^-}$, $B_{K^{-}}$ and $B(^{A}ZK^{-})$. These
values decrease with the increment of the imaginary potential
depth. The decreased values are about $1\sim 2$ MeV, if $V_0$
changes from $15$ MeV to 50 MeV. The single-particle $K^-$ binding
energies of 1s states are about $7\sim 12$ MeV larger than the
$K^-$ binding energies of 1s states, $B^{1s}_{K^{-}}$. On the
other hand, the single-particle $K^-$ binding energies of 1p
states are about $7\sim 10$ MeV larger than the $K^-$ binding
energies of 1p states, $B^{1p}_{K^{-}}$. Mare\v{s} \emph{et al.}
also predicted $B^{s,p}_{K^-}> B_{K^{-}}$ in their calculations,
they defined the difference as ``rearrangement energy", which
relates the polarization of the nuclear core by the $K^-$. The
rearrangement energies of 1s states decrease monotonically with
the nucleon number as a whole in our calculations.

The 1s state binding energies of $K^-$, $B^{1s}_{K^{-}}$, for the
listed nuclei are in the range of $B^{1s}_{K^-}= 73\sim 96$ MeV. And
the binding energies of $K^-$ for 1p states, $B^{1p}_{K^{-}}$, range
from 22 MeV to 63 MeV, increase monotonically with the nucleon
number $A$. Recently the FINUDA experiment predicted there is a
$K^-pp$ cluster with binding energy $B_{\mathrm{Kpp}}\sim 115$
MeV\cite{b4}, subtracting the binding energy between the two
protons, $B_{pp}=27.2$ MeV, the $K^-$ binding energy is $\sim 88$
MeV. Our predictions ($73\sim 96$ MeV ) are compatible with the
FINUDA experimental values  and the theoretical calculations in
\cite{a9,a10}.

The separations between the bound state 1p and 1s,
($B^{1s}_{K^{-}}-B^{1p}_{K^{-}}$), are on the order of $27\sim 62$
MeV, decrease with the increment of the nucleon number in general.
Increasing the imaginary potential depth from $V_{0}=15$ MeV to
$50$ MeV, the separations between the 1p and 1s states will
increase $\sim 1$ MeV.

On the other hand, from Tab.\ \ref{Calll}, we find that the widths
of the 1s kaonic bound states are about $12\pm 4$ MeV, $23\pm 9$
MeV and $42\pm 14$ MeV for $V_{0}=15$, $30$ and $50$ MeV,
respectively. The 1p state widths are on the order of 20, 40, for
$V_{0}=15$ and $30$ MeV, respectively. And for $V_{0}=50$ MeV, the
1p state widths are about $60\sim 80$ MeV, decrease with the
increment of the nucleon number in general. Mare\v{s} \emph{et
al.} analyzed the FINUDA experiment, considering the effects of
Fermi-motion on the decay widths the $K^-pp$ decay width $\Gamma
\sim 67$ MeV will be reduced down to 53 MeV. This result agrees
with our predictions $\Gamma^{1s}=42\pm 14$ MeV with  $V_{0}=50$
MeV.

From Tab.\ \ref{Calll}, we can also see that, if the depth
$V_{0}\leq 30$ MeV, the sum of the half widths of the 1s and 1p
states predicted by us are narrower than the separations between
$B^{1s}_{K^{-}}$ and $B^{1p}_{K^{-}}$, which implies that some
discrete states should be identified in experiment for these nuclei.
However, if $V_{0}\geq 50$ MeV the sum of the half widths of the 1s
and 1p states for all the listed nuclei, except $^{12}$C$K^-$,
predicted by us are larger than the separations of the two lowest
energy levels, thus no discrete states can be identified in
experiment.

\begin{widetext}
\begin{center}
\begin{table}[ht]
 \caption{The r.m.s. radii of neutron, proton and charge distributions, $r_n$, $r_p$
and $r_{ch}$ (in fm), respectively. $V_0$ (in MeV) is the
imaginary potential depth at normal nuclear density.} \label{Cal}
 \begin{tabular}{ccccccccccccccccccccc}  \hline \hline
\\
&$V_0$&$r_{p}$&$r_n$& $r_{ch}$  & & &$V_0$&$r_{p}$&$r_n$&
$r_{ch}$ & & &$V_0$&$r_{p}$&$r_n$& $r_{ch}$  \\
\hline \hline
 \\
$^{12}$C &         &\ \ 2.32 &\ \ 2.30  &\ \ 2.46               && $^{24}$Mg&         &\ \ 2.86 &\ \ 2.82 &\ \ 2.97          & &$^{36}$Ar &     &\ \ 3.26 &\ \ 3.21 &\ \ 3.36     \\
$^{12}$C$K^-$ & 15 &\ \ 2.20 &\ \ 2.18  &\ \ 2.35            && $^{24}$Mg$K^{-}$&15   &\ \ 2.80 &\ \ 2.77 &\ \ 2.92     & &$^{36}$Ar$K^{-}$ &15 &\ \ 3.22 &\ \ 3.17 &\ \ 3.32   \\
              & 30 &\ \ 2.20 &\ \ 2.18  &\ \ 2.35            &&                 &30   &\ \ 2.80 &\ \ 2.77 &\ \ 2.92                      & &&30 &\ \ 3.22 &\ \ 3.17 &\ \ 3.32 \\
              & 50 &\ \ 2.21 &\ \ 2.19  &\ \ 2.35            &&                 &50   &\ \ 2.80 &\ \ 2.77 &\ \ 2.92                      & &&50 &\ \ 3.22 &\ \ 3.18 &\ \ 3.33  \\
\\
$^{16}$O&          &2.58 &2.55 &2.70             &&$^{28}$Si&            &\ \ 2.93 &\ \  2.90  &\ \ 3.04       & &$^{40}$Ca &          &\ \ 3.36&\ \ 3.31&\ \ 3.46 \\
$^{16}$O$K^{-}$   &15&2.52 &2.49 &2.65             && $^{28}$Si$K^{-}$&15&\ \ 2.88 &\ \  2.85  &\ \ 2.99       & &$^{40}$Ca$K^{-}$&15  &\ \ 3.32&\ \ 3.27&\ \ 3.42    \\
                 &30&2.51 & 2.49 & 2.64           &&                 &30 &\ \ 2.88 &\ \  2.85  &\ \ 2.99         & &&30                &\ \ 3.32&\ \ 3.28&\ \ 3.42 \\
                 &50&2.51 & 2.48 & 2.64           &&                 &50 &\ \ 2.88 &\ \  2.85  &\ \ 2.99         & &&50                &\ \ 3.32&\ \ 3.28&\ \ 3.43  \\
\\
         $^{20}$Ne&&2.82&2.74&2.94               & & $^{32}$S &          &\ \ 3.13 &\ \ 3.09  &\ \ 3.24           & &$^{44}$Ti &         &\ \ 3.44 &\ \ 3.39 &\ \ 3.54     \\
$^{20}$Ne$K^{-}$&15&2.77&2.69&2.89               & & $^{32}$S$K^{-}$&15  &\ \ 3.08 &\ \ 3.04  &\ \ 3.19           & &$^{44}$Ti$K^{-}$&15 &\ \ 3.41 &\ \ 3.36 &\ \ 3.50      \\
              &30&2.77 & 2.68  & 2.89           & & &30                  &\ \ 3.08 &\ \ 3.04  &\ \ 3.19                         & &&30   &\ \ 3.41 &\ \ 3.36 &\ \ 3.50     \\
              &50& 2.76 & 2.68 & 2.88            & & &50                 &\ \ 3.09 &\ \ 3.04  &\ \ 3.20                      & &&50      &\ \ 3.41 &\ \ 3.36 &\ \ 3.51     \\
\\
 \hline \hline
\end{tabular}
\end{table}
\end{center}
\end{widetext}

\subsection{The static properties of the kaonic nuclei}

First, let's see Tab.\ \ref{Cal}. We find that the  r.m.s. radii of
neutron, proton and charge distributions, $r_n$, $r_p$ and $r_{ch}$
for the kaonic nuclei, $^AZK^-$, are smaller than those for the
corresponding ordinary nuclei, $^AZ$. For example, the r.m.s.\ radii
of $^{12}$C$K^-$ reduce about 0.12 fm, when a $K^-$-meson is
injected into $^{12}$C. For the other heavier kaonic nuclei, the
decreased values of the r.m.s. radii are about $0.04\sim 0.06$ fm.
Generally, the r.m.s. radii of lighter kaonic nuclei decrease more
obviously than those of heavier kaonic nuclei. The phenomenon of the
r.m.s.\ radii decreased because of a meson or hyperon being bound in
a nucleus is called ``shrinkage'', which has been found in some
lighter $\Lambda$ hypernuclei \cite{e1} in experiment. Within the
framework of RMF, we also find the shrinkage effects exist in
$\Lambda$-, $\Theta^+$- hypernuclei \cite{e2,e3}.

From the table, we also find that the r.m.s.\ radii do not change
obviously with the imaginary potential depth. If there are
differences between the r.m.s.\ radii for different imaginary
potential depth, the differences are within 0.01 fm.

To explain the ``shrinkage effect'', we should start with the $K^-$
properties in nuclear matter and the equations of motion for kaonic
nuclei in RMF theory. Firstly, the $K^-$ meson does not identify
with nucleons and can enter the nuclear cores. Secondly, the strong
$K^-N$ attraction can produce a strong $K$ field in nuclei, which
increases the strength of the scalar field $\sigma$\ (see Eq.\
(\ref{eossimp2})) and decreases the vector field $\omega$\ [see Eq.\
(\ref{eossimp3})]. This means the attraction between nucleons become
larger and the repulse between the nucleons become weaker. Thus, the
nucleons in the nucleus are bounded more tightly, which results in
the so called ``shrinkage effect".

\begin{widetext}
\begin{center}
\begin{figure}[ht]
\centering \epsfxsize=12 cm \epsfbox{ddd.eps}
\caption{\footnotesize Nucleon-density as a function of nucleus
radius. The solid and dotted curves are for kaonic nuclei and
corresponding ordinary nuclei, respectively.}\label{Dens}
\end{figure}
\end{center}
\end{widetext}

\subsection{nuclear density of the kaonic nuclei}

The distributions of the nuclear density for some light and moderate
kaonic nuclei in their ground states together with those of
corresponding ordinary nuclei are shown in Fig.\ \ref{Dens}. In the
calculations, we set $V_0=30$ MeV.

From the figure we can see, in the interior of the kaonic nuclei,
the nuclear densities become much denser than those of the ordinary
nuclei. The central nuclear density is enhanced by a value $0.03\sim
0.1$ fm$^{-3}$ for all the listed kaonic nuclei. In all the kaonic
nuclei,  the lowest enhanced central nuclear density is $\sim 0.03$
fm$^{-3}$ for $^{44}$Ti$K^-$, and the highest enhanced central
nuclear density is $\sim 0.1$ fm$^{-3}$ for $^{12}$C$K^-$. In all
the listed kaonic nuclei, the central nuclear density of
$^{12}$C$K^-$ is the densest, $\sim 0.31 fm^{-3}$, which is around
2.1 times the normal nuclear density $\rho_{0}$.

Our calculations give obvious enhancement of the nuclear density in
the interior of the kaonic nuclei.
However, this value is much less than the prediction in  \cite{a9}.
In \cite{a9}, Akaishi \emph{et al.} gave the central density about 3
times that of the $\alpha$ particle for $^8$Be$K^-$ system, and in
\cite{a10}, the central nuclear density of kaonic nuclei gave by
Dot\'{e} \emph{et al.} even reaches to $0.71\sim1.50$ fm$^{-3}$,
which is about $4\sim10$ times the normal nuclear density with a
framework of antisymmetrized molecular dynamics. It is indeed an
amazing high central density $\rho(0)=0.81$ fm$^{-3}$$\sim5\rho_{0}$
for the many-body system $^{11}$C$K^-$ in \cite{a10} compared with
our results.

\section{Summary}

We study the kaonic nuclei within the framework of RMF theory. All
the equations are solved self-consistently. We carry out systematic
calculations for the light and moderate possible kaonic nuclei, from
$^{12}$C to $^{44}$Ti. The energy spectra of these kaonic nuclei are
studied, the ground state (1s state) binding energies of $K^-$  are
obtained, which are in the range of $73\sim 96$ MeV for all the
possible kaonic nuclei studied. The binding energies of of $K^-$ for
1p states are in the range of $22\sim 63$ MeV, and increase
monotonically with the nucleon number $A$. The separations between
the bound state 1p and 1s are in the range of $27\sim 62$ MeV, and
decrease with the increment of the nucleon number generally.

The upper limit widths of the 1s and 1p states are $42\pm 14$ MeV
and $71\pm 10$ MeV, respectively, with $V_{0}\sim 50$ MeV. And the
lower limit widths of the 1s and 1p states are $12\pm 4$ MeV and
$21\pm 3$ MeV, respectively, with $V_{0}\sim 15$ MeV.

The binding energies of $K^-$, $B_K$, will decrease $1\sim 2$ MeV,
if we increase the imaginary potential depths in their possible
range. The widths do not affect the r.m.s. radii obviously.

If the imaginary potential depth $V_{0}$ is deeper than $50$ MeV,
the sum of the half widths of the 1s and 1p states predicted by us
for $^{16}$O$K^{-}$, $^{20}$Ne$K^{-}$, $^{24}$Mg$K^{-}$,
$^{28}$Si$K^{-}$, $^{32}$S$K^{-} $, $^{36}$Ar$K^{-}$,
$^{40}$Ca$K^{-}$ and $^{44}$Ti$K^{-}$ are larger than the
separations between the 1s and 1p states for these kaonic nuclei,
which implies that no discrete states can be identified in
experiment for these nuclei. However, if $V_{0}\leq 30$ MeV the sum
of the half widths of the 1s and 1p states predicted by us are
narrower than the separations of the two lowest energy levels, thus,
the discrete $K^-$ bound states should be identified in experiment.

The shrinkage effects for kaonic nuclei are found. The interior
nucleon-densities do not increase drastically. The densest central
nuclear density predict by us is about 2.1 times the normal nuclear
density, however, which is much less than the  value predicted in
\cite{a9}.

\section*{Acknowledgements}
We would like to thank Prof. A. Gal for helpful discussions. This
work was supported in part by the Natural Science Foundation of
China (10275037, 10375074, 90203004) and China Doctoral Programme
Foundation of Institution of Higher Education (20010055012).

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text:
\section{Introduction}

We continue the study of non-Hamiltonian graphs
with the property that removing an arbitrary set of vertices
disconnects the graph into a relatively small number of components (compared to the size of the removed set).
In the present paper, we construct families of maximal planar such graphs whose longest cycles are
short (compared to the order of the graph).

More formally, the properties which we study are
the toughness of graphs and the shortness exponent of classes of graphs
(both introduced in 1973).
We recall that following Chv{\'a}tal~\cite{chva}, the \emph{toughness} of a
graph $G$ is the minimum, taken over all separating sets $X$ of
vertices in $G$, of the ratio of $|X|$ to $c(G - X)$
where $c(G - X)$ denotes the number of components of the graph $G - X$.
The toughness of a complete graph is defined as being infinite.
We say that a graph is \emph{$t$-tough} if its toughness is at least $t$.

Along with the definition of toughness,
Chv{\'a}tal~\cite{chva} conjectured that there is a constant $t_0$
such that every $t_0$-tough graph
(on at least three vertices) is Hamiltonian.
As a lower bound on $t_0$,
Bauer et al.~\cite{74} presented
graphs with toughness arbitrarily close to $\frac94$
which contain no Hamilton path (and thus, they are non-Hamiltonian).
While remaining open for general graphs, Chv\'{a}tal's conjecture was
confirmed in several restricted classes of graphs; and also various
relations among the toughness of a graph and properties of its cycles are known.
We refer the reader to the extensive survey on this topic~\cite{surv}.

Clearly,
every graph (on at least five vertices) of toughness greater than $\frac{3}{2}$ is $4$-connected,
so every such planar graph is Hamiltonian
by the classical result of Tutte~\cite{Tutte}.
On the other hand, Harant~\cite{hara} showed that not every $\frac{3}{2}$-tough planar graph is
Hamiltonian; and furthermore,
the shortness exponent of the class of
$\frac{3}{2}$-tough planar graphs is less than $1$.

We recall that following Gr{\"u}nbaum and Walther~\cite{sef},
the \emph{shortness exponent} of a
class of graphs $\Gamma$ is the $\liminf$,
taken over all infinite sequences $G_n$ of non-isomorphic graphs of $\Gamma$
(for $n$ going to infinity),
of the logarithm of the length of a longest cycle in $G_n$
to base equal to the order of $G_n$.

Introducing this notation, Gr{\"u}nbaum and Walther~\cite{sef}
also presented upper bounds on the shortness exponent
for numerous subclasses of the class of $3$-connected planar graphs.
Furthermore, they remarked that the upper bound for the class of $3$-connected planar graphs itself
was presented earlier by Moser and Moon~\cite{momo} who used a slightly different notation.
Later, Chen and Yu~\cite{3-conn} showed that
every $3$-connected planar graph $G$
contains a cycle of length at least $|V(G)|^{\log_3 2}$;
in combination with the bound of~\cite{momo},
it follows that the shortness exponent of this class equals $\log_3 2$.
A number of results considering the shortness exponent and similar parameters
are surveyed in~\cite{owens}.

Considering the class of maximal planar graphs under a certain toughness restriction,
Owens~\cite{owensTatra} presented non-Hamiltonian maximal planar graphs of
toughness arbitrarily close to $\frac{3}{2}$.
Harant and Owens~\cite{5/4} argued that the shortness exponent of the class
of $\frac{5}{4}$-tough maximal planar graphs is at most $\log_9 8$.
Improving the bound $\log_7 6$ presented by Dillencourt~\cite{dill},
Tk\'{a}\v{c}~\cite{tkac} showed that it is at most $\log_6 5$
for the class of $1$-tough maximal planar graphs.

In the present paper, we show the following.

\begin{theorem}\label{main}
Let $\sigma$ be the shortness exponent of the class of maximal planar graphs
under a certain toughness restriction.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item If the graphs are $\tfrac{5}{4}$-tough, then $\sigma$ is at most $\log_{30}{22}$.
\item If the graphs are $\tfrac{8}{7}$-tough, then $\sigma$ is at most $\log_{6}{5}$.
\item If the toughness of the graphs is greater than $1$, then $\sigma$ equals $\log_{3}{2}$.
\end{enumerate}
\end{theorem}

We note that $\log_{9}{8} > \log_{30}{22}$,
that is, the statement in item $(i)$ of Theorem~\ref{main} improves the result of~\cite{5/4}.
Furthermore,
items $(ii)$ and $(iii)$
provide two different generalizations of the result of~\cite{tkac}
since $\frac{8}{7} > 1$ and $\log_{6}{5} > \log_{3}{2}$.

We remark that we fix a problem in a technical lemma presented in~\cite[Lemma~1]{5/4}.
The fixed version of this lemma (see Lemma~\ref{constr2}) is applied to prove the present results.

\section{Structure of the proof}
\label{s2}

In order to prove Theorem~\ref{main},
we shall construct three families of graphs whose properties are summarized in the following proposition.

\begin{prop}\label{shortAll}
For every $i = 1,2,3$ and
every non-negative integer $n$,
there exists a maximal planar graph $F_{i,n}$ on $f_i(n)$ vertices
whose longest cycle has
$c_i(n)$ vertices where
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item $f_1(n) = 1 + 101(1 + 30 + \dots + 30^n)$ and $c_1(n) = 1 + 93(1 + 22 + \dots + 22^n)$
and $F_{1,n}$ is $\frac{5}{4}$-tough,
\item $f_2(n) = 1 + 14(1 + 6 + \dots + 6^n)$ and $c_2(n) = 1 + 13(1 + 5 + \dots + 5^n)$
and $F_{2,n}$ is $\frac{8}{7}$-tough,
\item $f_3(n) = 4 + 5(1 + 3 + \dots + 3^n)$ and $c_3(n) = 3 \cdot 2^{n+3}-9n-15$
and the toughness of $F_{3,n}$ is greater than $1$.
\end{enumerate}
\end{prop}

Before constructing the graphs $F_{1,n}$,
we point out that the use of Proposition~\ref{shortAll}
leads directly to
the main results of the present paper.
\begin{proof}[Proof of Theorem~\ref{main}]
We consider an infinite sequence of non-isomorphic graphs $F_{1,n}$
given by item $(i)$ of Proposition~\ref{shortAll};
and we recall that they are $\frac{5}{4}$-tough maximal planar graphs.
Furthermore, we have
\begin{equation*}
f_1(n) = 1 + \tfrac{101}{29}(30^{n+1} - 1)
\quad
\textnormal{and}
\quad
c_1(n) = 1 + \tfrac{93}{21}(22^{n+1} - 1).
\end{equation*}
It follows that
\begin{equation*}
\lim_{n\to\infty} \log_{f_1(n)} c_1(n) = \log_{30}{22}.
\end{equation*}
Thus, the considered shortness exponent is at most $\log_{30}{22}$.

Using similar arguments and considering items $(ii)$ and $(iii)$ of Proposition~\ref{shortAll},
we obtain the desired upper bounds.

Clearly, if $G$ is a maximal planar graph (on at least four vertices), then it is $3$-connected.
By a result of~\cite{3-conn}, $G$ contains a cycle of length at least $|V(G)|^{\log_3 2}$.
In combination with the upper bound obtained due to item
$(iii)$ of Proposition~\ref{shortAll}, we obtain that
for the class of maximal planar graphs of toughness greater than $1$,
the shortness exponent equals $\log_3 2$.
\end{proof}

In the remainder of the present paper, we construct the families of graphs
having the properties described in Proposition~\ref{shortAll}.
Basically, we proceed in four steps.
First, we introduce relatively small graphs $F_{i, 0}$ called `building blocks' in Section~\ref{sb},
and we observe key properties of their longest cycles.
We use these building blocks to construct larger graphs $F_{i, n}$ in Section~\ref{sc},
and we show that their longest cycles are short.
In Section~\ref{tb}, we study the toughness of the building blocks.
The toughness of the graphs $F_{i, n}$ is shown
in Sections~\ref{sg} and~\ref{tt}.

We remark that the graphs
$F_{1, n}$ and $F_{2, n}$
are obtained using a standard construction for bounding the shortness exponent
(see for instance~\cite{sef}, \cite{dill}, \cite{5/4}, \cite{tkac} or~\cite{chpl});
the improvement of the known bounds comes with the choice of suitable building blocks.
In addition, we formalize the key ideas of this construction to make them more accessible for further usage.

The construction of graphs $F_{3, n}$
can be viewed as a simple modification of the construction used in~\cite{momo}
(yet the toughness and longest cycles of the constructed graphs are different).

\section{Building blocks}
\label{sb}

We start by considering the graph $T$ depicted in Figure~\ref{f:54}
which plays an important role in the latter constructions.
We let $o_1$, $o_2$, and $o_3$ be its vertices of degree $6$.

A \emph{$T$-region} of a graph $G$ is an induced subgraph isomorphic to $T$
with a distinction of vertices referring to $o_1$, $o_2$, $o_3$ as to \emph{outer} vertices
and to the remaining vertices as to \emph{inner} vertices,
and with the property that no inner vertex is adjacent to a vertex of $G - T$.
Similarly, we define an \emph{$H$-region} for a given graph $H$ and a given distinction of its vertices.

We view the $T$-regions as replacing triangles of a graph with copies of $T$ in the natural way.
The basic idea of the present constructions is that if these triangles share many vertices,
then every cycle in the resulting graph misses many vertices.
We formalize this idea in Proposition~\ref{T}.

We recall that a vertex is called \emph{simplicial} if its neighbourhood induces a complete graph.

\begin{prop}\label{T}
Let $R$ be a $T$-region of a graph $G$ and let $C$ be a cycle containing
all three simplicial vertices of $R$ and a vertex of $G - R$.
Then the subgraph of $C$ induced by the vertices of $R$ is a path containing all outer vertices of $R$;
two of them as its ends.
\end{prop}
\begin{proof}
Clearly, $R$ has three simplicial vertices none of which is an outer vertex.
The statement follows from the fact that every simplicial vertex has only one neighbour
which is not an outer vertex.
\end{proof}

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.6]{54.pdf}
    \caption{The graph $T$ and the construction of the graph $F_{1,0}$.
    The graph $F_{1,0}$ is obtained by replacing each of the highlighted triangles
    (of the graph on the left)
    with a copy of $T$ in the natural way
    (by identifying the vertices of the highlighted triangle
    with vertices $o_1, o_2, o_3$ of $T$).}
    \label{f:54}
\end{figure}

Aiming for the graph $F_{1,0}$ (the building block for constructing the graphs $F_{1,n}$),
we consider a graph with a number, say $r$, of $T$-regions
which all share one outer vertex and which are otherwise disjoint.
By Proposition~\ref{T}, every cycle in this graph contains at most $2r + 2$
of the $3r$ simplicial vertices belonging to these $T$-regions.
(We view the building block used in~\cite{5/4} simply as choosing $r = 3$.)
With hindsight, we remark that the used construction leads to
the upper bound $\log_{3r}{(2r+2)}$ on the shortness exponent;
so we minimize this function over all integers $r \geq 3$,
that is, we choose $r = 10$.

We let $F_{1,0}$ be the graph depicted in Figure~\ref{f:54},
and we note that $F_{1,0}$ is a maximal planar graph.
Clearly, $F_{1,0}$ has $30$ simplicial vertices, and we colour these vertices white.
We recall that every cycle in $F_{1,0}$ contains at most $22$ white vertices.

Furthermore, we let $F_{2, 0}$ be the graph depicted in Figure~\ref{f:87}.
Clearly, $F_{2, 0}$ is a maximal planar graph having $6$ simplicial vertices;
and we colour these vertices white.
We use Proposition~\ref{T}, and we observe that a cycle in $F_{2, 0}$
contains at most $5$ white vertices.
Lastly, we let $F_{3,0}$ be the graph $T$.

For every $i = 1,2,3$, we define the outer face of $F_{i,0}$
as given by the present embedding (see Figures~\ref{f:54} and~\ref{f:87}).
In the following section, we shall use the blocks $F_{i,0}$ and construct the graphs $F_{i,n}$.

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.6]{87.pdf}
    \caption{The construction of the graph $F_{2, 0}$.
    The highlighted triangle represents a subgraph $T$.}
    \label{f:87}
\end{figure}

\section{Families of tree-like structured graphs}
\label{sc}

We recall the standard construction used for bounding the shortness exponent,
and we formalize it with the following definition and Lemma~\ref{cyc}.

An \emph{arranged block} is a $5$-tuple $(G_0, j, W, O, k)$
where $G_0$ is a graph, $j$ is the number of vertices of $G_0$,
and $W$ and $O$ are disjoint sets of vertices of $G_0$
such that the vertices of $W$ are simplicial and independent
and $O$ induces a complete graph
and such that every cycle in $G_0$ contains at most $k$ vertices of $W$.

\begin{lemma}\label{cyc}
Let $(G_0, j, W, O, k)$ be an arranged block such that $k \geq 1$.
For every $n \geq 1$, let $G_n$ be a graph obtained from $G_{n-1}$
by replacing every vertex of $W$ with a copy of $G_0$ (which contains $W$ and $O$),
and by adding arbitrary edges which connect the neighbourhood of the replaced vertex
with the set $O$ of the copy of $G_0$ replacing this vertex.
Then $G_n$ has $1 + (j - 1)(1 + |W| + \dots + |W|^n)$ vertices and its longest cycle has at most
$1 + (\ell - 1)(1 + k + \dots + k^n)$
vertices where $\ell = j - |W| + k$.
\end{lemma}
\begin{proof}
We note that $G_n$ contains $|W|^{n+1}$ vertices of $W$.
For the sake of induction, we prove the statement with an additional claim
that every cycle in $G_n$ contains at most $k^{n+1}$ vertices of $W$.
Clearly, the statement and the claim are satisfied for $n = 0$, and we proceed by induction on $n$.

We note that the difference in the order of $G_n$ and $G_{n-1}$ equals $(j - 1)|W|^n$.
Thus, $G_n$ has $1 + (j - 1)(1 + |W| + \dots + |W|^n)$ vertices.

We let $C$ be a cycle in $G_n$, and we view this cycle simply as a sequence of vertices.
For every newly added copy of $G_0$, we remove from $C$ all but one vertex of the copy
and we replace the remaining vertex (if there is such) by the corresponding replaced vertex of $G_{n-1}$;
and we let $C'$ denote the resulting sequence.
Clearly, if $C'$ has at most two vertices, then $C$ visits at most one of the newly added copies of $G_0$.
If $C'$ has at least three vertices,
then $C'$ defines a cycle in $G_{n-1}$
(since the neighbourhood of every vertex of $W$ in $G_{n-1}$ induces a complete graph);
and $C'$ contains at most $k^n$ vertices of $W$ (by the induction hypothesis).
Thus, $C$ visits at most $k^n$ of the newly added copies of $G_0$.

Similarly, we choose an arbitrary newly added copy of $G_0$, and
we remove from $C$ all vertices not belonging to this copy.
We note that the resulting sequence either contains at most two vertices (belonging to $O$)
or it is a cycle in $G_0$ (since $O$ induces a complete graph).
Thus, a cycle in $G_n$ contains at most $k$ vertices belonging to $W$ of one copy of $G_0$.
Furthermore, a cycle contains at most $j - |W| + k$ vertices of one such copy.

Consequently, $C$ contains at most $k^{n+1}$ vertices of $W$.
Furthermore, the length of $C$ minus the length of a longest cycle in $G_{n-1}$
is at most $(j - |W| + k -1)k^n$ which concludes the proof.
\end{proof}

For $i = 1,2$, we consider this construction for the graph $F_{i,0}$ playing the role of $G_0$
and the set of its white vertices playing the role of $W$
and the set of vertices of its outer face playing the role of $O$.
For every added copy of $F_{i,0}$, we join the vertices of its outer face
to the neighbourhood of the corresponding replaced vertex
by adding six edges in such a way that the new edges form a $2$-regular bipartite graph
(that is, a cycle of length $6$).
We let $F_{i, n}$ be the resulting graphs, and we observe that they are maximal planar graphs.
For instance, see the graph $F_{2, 1}$ depicted in Figure~\ref{f:F21}.

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.6]{F21.pdf}
    \caption{The construction of the graph $F_{2, 1}$.
    The graph $F_{2, 1}$ is obtained from the smaller graph (left) by replacing each of its highlighted triangles
    with the larger graph (right) in the natural way.
    (This corresponds to replacing white vertices of $F_{2, 0}$ with copies of $F_{2, 0}$
    and adding edges as in the present construction.)}
    \label{f:F21}
\end{figure}
We start verifying that the constructed graphs (for $i = 1,2$) have the desired properties.

\begin{coro}\label{cyc123}
For every $i = 1,2$ and every non-negative integer $n$,
the graph $F_{i,n}$ has $f_i(n)$ vertices
and its longest cycle has
$c_i(n)$ vertices where
\begin{enumerate}
\item $f_1(n) = 1 + 101(1 + 30 + \dots + 30^n)$ and $c_1(n) = 1 + 93(1 + 22 + \dots + 22^n)$,
\item $f_2(n) = 1 + 14(1 + 6 + \dots + 6^n)$ and $c_2(n) = 1 + 13(1 + 5 + \dots + 5^n)$.
\end{enumerate}
\end{coro}
\begin{proof}
The order of the graphs
and the upper bound on the length of their longest cycles follow from Lemma~\ref{cyc}.

We note that a longest cycle in $F_{1,0}$, $F_{2,0}$ has $94$, $14$ vertices, respectively.
Furthermore, there is a longest cycle which contains an edge of the outer face.
Clearly, by removing this edge from the cycle we obtain a path whose ends are vertices of the outer face.
We consider a longest cycle in $F_{i,n-1}$ and we extend it to a cycle in $F_{i,n}$ using these paths.
The following observation shows that such an extension is possible.
For an arbitrary pair, say $A$, of neighbours of one replaced vertex
and an arbitrary pair, say $B$, of vertices of the outer face of the corresponding $F_{i,0}$ (used for replacing this vertex),
the bipartite graph $(A, B)$ has a perfect matching.

A simple counting argument gives that $F_{i,n}$ contains a cycle of length $c_i(n)$,
for every $i = 1,2$ and every $n$.
\end{proof}

We use a slightly different construction to get the graphs $F_{3, n}$.
The building block $F_{3,0}$ is the graph $T$ whose simplicial vertices are coloured white.
We view a subgraph induced by a white vertex and its neighbourhood as a
\emph{$K_4$-region}, that is, a subgraph isomorphic to $K_4$
whose white vertex has degree $3$ in the whole graph, and the neighbours of
the white vertex are called \emph{outer} vertices of the $K_4$-region.

For $n \geq 1$, we let $F_{3, n}$ be a graph obtained from $F_{3, n-1}$
by replacing every $K_4$-region of $F_{3, n-1}$ with a $T$-region
in the natural way (the outer vertices of
the $K_4$-region are the outer vertices of
the $T$-region);
and we note that they are maximal planar graphs.
For instance, the graph $F_{3, 1}$ is depicted in Figure~\ref{f:F31}.
We proceed by the following proposition.

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.6]{F31.pdf}
    \caption{The construction of the graph $F_{3, 1}$.
    Each highlighted triangle represents a subgraph $T$.}
    \label{f:F31}
\end{figure}

\begin{prop}\label{cyc4}
For every non-negative integer $n$, the graph $F_{3,n}$
has $4 + 5(1 + 3 + \dots + 3^n)$ vertices
and its longest cycle has $3 \cdot 2^{n+3}-9n-15$ vertices.
\end{prop}
\begin{proof}
We verify the order of $F_{3,n}$ using induction on $n$.
Clearly, $F_{3,0}$ has $9$ vertices,
and we note that the difference in the order of $F_{3,n}$ and $F_{3,n-1}$ equals $5 \cdot 3^n$.
Thus, $F_{3,n}$ has $4 + 5(1 + 3 + \dots + 3^n)$ vertices.

We show the length of a longest cycle using a slightly technical argument.
We let $s_i(n)$ denote the length of a longest cycle in $F_{3,n}$
which contains
$i$ edges of the outer face
(in the embedding which follows naturally from the construction).
For the sake of induction, we prove the following equalities:
\begin{align}
\label{eq:1}
s_0(n) & = 3s_1(n-1) -3 \nonumber \\
s_1(n) & = 2s_2(n-1) + s_1(n-1) -3 \\
s_2(n) & = 2s_2(n-1) \nonumber
\end{align}
and
\begin{align}
\label{eq:2}
s_0(n) & = 3 \cdot 2^{n+3}-9n -15 \nonumber\\
s_1(n) & = 2^{n+4} - 3n - 7 \\
s_2(n) & = 2^{n+3}. \nonumber
\end{align}

Clearly, $s_0(0) = s_1(0) = 9$ and $s_2(0) = 8$,
and (using Proposition~\ref{T}) we note that $s_0(1) = 24$, $s_1(1) = 22$ and $s_2(1) = 16$;
so the equalities are satisfied for $n = 1$.
We assume that they are satisfied for $n-1$ and we prove them for $n$.

For $n \geq 1$, we view $F_{3,n}$ as the graph
obtained from $F_{3,0}$ by replacing each of its
$K_4$-regions with an $F_{3,n-1}$-region;
and we view a cycle, say $C$, of $F_{3,n}$ as a sequence of vertices.
We consider one of the $F_{3,n-1}$-regions, say $R$,
and we remove from $C$ all vertices not belonging to $R$;
and we let $C'$ be the resulting sequence.
We observe that $C'$ either is a cycle or it has at most two vertices.
Furthermore, if $C$ visits a vertex not belonging to $R$,
then $C'$ (if it has at least three vertices)
is a cycle in $R$ containing at least one edge of its outer face.

Clearly, a longest cycle (in $F_{3,n}$) whose all vertices belong to the same $F_{3,n-1}$-region
has $s_0(n-1)$ vertices, and we observe that a longest cycle
visiting more than one of the regions has $3s_1(n-1) -3$ vertices.
We use~\eqref{eq:2} for $s_0(n-1)$ and $s_1(n-1)$,
and we note that
\begin{align*}
3 \cdot 2^{n+2}-9(n-1)-15 < 3(2^{n+3}- 3(n-1) - 7) -3.
\end{align*}
So we get
$s_0(n) = 3s_1(n-1) -3$.
By similar arguments, we get
$s_1(n) = 2s_2(n-1) + s_1(n-1) -3$
and
$s_2(n) = 2s_2(n-1)$.

Consequently, we can use~\eqref{eq:1} for $s_i(n)$, and we obtain
\begin{align*}
s_0(n) =  3s_1(n-1)-3 =  3(2^{n+3} - 3(n-1) - 7) - 3 = 3 \cdot 2^{n+3}-9n-15
\end{align*}
and
\begin{align*}
s_1(n) &= 2s_2(n-1) + s_1(n-1) -3 = 2 \cdot 2^{n+2} + 2^{n+3} - 3(n-1) - 7 - 3 \\
&= 2^{n+4} - 3n - 7
\end{align*}
and
\begin{align*}
s_2(n) = 2s_2(n-1) = 2 \cdot 2^{n+2} = 2^{n+3}.
\end{align*}
Thus, the equalities are satisfied for $n$.
In particular, a longest cycle in $F_{3,n}$ has $3 \cdot 2^{n+3}-9n-15$ vertices.
\end{proof}

With Corollary~\ref{cyc123} and Proposition~\ref{cyc4} on hand,
we shall focus on the toughness of the constructed graphs.

\section{Toughness of the extended blocks}
\label{tb}

In this section, we study the toughness of $F_{i,0}$ (for $i = 1,2,3$)
and of its extension $F^+_{i,0}$ (for $i = 1,2$) which is
a graph obtained by adding a vertex adjacent to all vertices of the outer face
of $F_{i,0}$.
We shall use the following two propositions.

\begin{prop}\label{simpl}
Adding a simplicial vertex to a graph does not increase its toughness.
\end{prop}
\begin{proof}
We let $x$ be a simplicial vertex of a graph $G^+$ and we let $G = G^+ - x$
and we let $S$ be a set of vertices of $G$.
We note that
$c(G^+-S) \geq  c(G-S)$,
and the statement follows.
\end{proof}

\begin{prop}\label{Ttough}
Let $R$ be a $T$-region of a graph $G$
and let $I$, $O$ be the set of all inner, outer vertices of $R$, respectively.
Let $t \geq 1$
and let $S$ be a set of vertices of $G$
such that $c(G - S) > \frac{1}{t}|S|$.
If $|S \cap O| \geq 2$,
then there is a separating set $S' = (S \setminus I) \cup A$
such that $c(G - S') > \frac{1}{t}|S'|$
where $A$ is chosen as follows.

\begin{itemize}
\item If $|S \cap O| = 2$, then $A$ consists of
the non-simplicial vertex of $I$ which is the common neighbour
of the vertices of $S \cap O$.
\item If $|S \cap O| = 3$,
then $A$ consists of two non-simplicial vertices of $I$.
\end{itemize}
\end{prop}
\begin{proof}
In both cases, we modify $S$ as suggested; and we let $S'$ be the resulting set.
Clearly, $S'$ is a separating set and $c(G-S') - c(G-S) \geq 0$, and we observe that
\begin{equation*}
c(G-S') - c(G-S) \geq |S'|- |S|.
\end{equation*}
Since $t \geq 1$, we have either $0 > \frac{1}{t}(|S'|- |S|)$ or $|S'|- |S| \geq \frac{1}{t}(|S'|- |S|)$.
Consequently, we obtain
\begin{equation*}
c(G-S') - c(G-S) \geq \tfrac{1}{t}|S'|- \tfrac{1}{t}|S|.
\end{equation*}
We use $c(G - S) > \frac{1}{t}|S|$ and we conclude that $c(G - S') > \frac{1}{t}|S'|$.
\end{proof}

We recall that a set $D$ of vertices is \emph{dominating} a graph $G$
if every vertex of $G - D$ is adjacent to a vertex of $D$.
We note that every pair of vertices of degree $6$ is dominating $T$,
so every separating set in $T$ has at least two of these vertices.
As a consequence of Proposition~\ref{Ttough}, we note the following.

\begin{coro}\label{32tough}
The graph $T$ is $\frac{3}{2}$-tough.
\end{coro}

Furthermore, if a graph contains a $T$-region
and a vertex not belonging to the $T$-region, then the toughness of the graph is at most $\frac{5}{4}$;
and we show that this is the correct value for $F^+_{1, 0}$ and $F_{1, 0}$.

\begin{prop}\label{54tough}
The graphs $F^+_{1, 0}$ and $F_{1, 0}$ are $\frac{5}{4}$-tough.
\end{prop}
\begin{proof}
By Proposition~\ref{simpl}, it suffices to show that $F^+_{1, 0}$ is $\frac{5}{4}$-tough.
To the contrary, we suppose that
there is a separating set $S$ of vertices
such that $c(F^+_{1, 0} - S) > \frac{4}{5}|S|$.
We consider such $S$ adjusted by using Proposition~\ref{Ttough},
in sequence, for all $T$-regions of $F^+_{1, 0}$.

We let $\mathcal I$ denote the set of all components of $F^+_{1, 0} - S$
consisting exclusively of inner vertices of some $T$-region.
We note that the existence of such component implies that
at least two outer vertices of the corresponding $T$-region belong to $S$
(since every pair of outer vertices of $T$ is dominating $T$).
We let $r_2$, $r_3$ denote the number of $T$-regions whose exactly $2$, $3$
outer vertices belong to $S$, respectively.
We let $c$ denote the common vertex of all $T$-regions.
Considering the inner vertices of the $T$-regions,
we call simplicial such vertices \emph{white}
and the remaining such vertices \emph{grey}.
Except for $c$, the outer vertices of the $T$-regions are called \emph{black}.
The vertices adjacent to a black vertex
but not belonging to a $T$-region are called \emph{blue}.
We let $c'$ denote the vertex adjacent to all blue vertices
and $x$ denote the vertex of $F^+_{1, 0}$ not belonging to $F_{1, 0}$.
We let $\mathcal B$ denote the set of all components of $F^+_{1, 0} - S$
containing a black vertex or a blue vertex.

We shall use a discharging argument to avoid complicated inequalities.
We assign charge $5$ to every component of $F^+_{1, 0} - S$,
and we aim to distribute all assigned charge among the vertices of $S$,
and to show that every vertex of $S$ receives charge at most $4$,
contradicting the assumption that $c(F^+_{1, 0} - S) > \frac{4}{5}|S|$.
We pre-distribute the charge according to the following rules.
\begin{itemize}
\item Every grey vertex of $S$ receives $4$
of the total charge of the components of $\mathcal I$ belonging to the same $T$-region as the grey vertex.
\item For every $T$-region, the remaining charge
of all components of $\mathcal I$ belonging to this $T$-region is distributed
equally among the black vertices of $S$ belonging to this $T$-region.
\item Every blue vertex of $S$ receives
as much of the total charge of components outside $\mathcal I$ as possible (at most $4$).
\end{itemize}
We note that after the pre-distribution, the charge of every component of $\mathcal I$ is $0$;
and we focus on the remaining charge of the rest of the components.

If $c(F^+_{1, 0} - S) - |\mathcal I| \leq 1$, then we have $|\mathcal I| \geq 1$ and the remaining charge is at most $5$.
Consequently, $r_2 \geq 1$ or $r_3 \geq 1$,
and in both cases, the vertices of $S$ can still receive charge at least $5$, a contradiction.

We assume that $c(F^+_{1, 0} - S) - |\mathcal I| \geq 2$.
We show that $F^+_{1, 0} - S$ has no component containing $c$ and no black vertex.
The existence of such component implies that
all black vertices belong to $S$, and these vertices can still receive $\frac{7}{2} \cdot 20$.
A contradiction follows by counting the maximum possible number of components not belonging to $\mathcal I$.

Consequently, $F^+_{1, 0} - S$ has at most one component belonging to neither $\mathcal I$ nor $\mathcal B$.
If $|\mathcal B| \leq 1$, then there is such component (since $c(F^+_{1, 0} - S) - |\mathcal I| \geq 2$).
Clearly, this component contains $c'$ or $x$, that is,
all blue vertices or $c'$ and at least two blue vertices belong to $S$,
and a contradiction follows.

We assume that $|\mathcal B| \geq 2$.
On the other hand, considering the graph induced by black and blue vertices,
we observe that the size of a maximum independent set
of this graph is $13$. Thus, $|\mathcal B| \leq 13$.

Since $|\mathcal B| \geq 2$, we note that at least $|\mathcal B|$ black and at least $|\mathcal B|$ blue vertices belong to $S$.
We let $d$ denote the number of blue vertices of $S$ minus $|\mathcal B|$.
We recall that $F^+_{1, 0} - S$ has at most $|\mathcal B| + 1$ components not belonging to $\mathcal I$.
Thus, the remaining charge, which is yet to be distributed, is at most $|\mathcal B| + 5 - 4d$;
and since $d \geq 0$, it is at most $|\mathcal B| + 5$.

If $c$ does not belong to $S$, then
the black vertices of $S$ can still receive at least $\frac{7}{2}|\mathcal B|$.
Clearly, $\frac{7}{2}|\mathcal B| \geq |\mathcal B| + 5$ since $|\mathcal B| \geq 2$, a contradiction.

We assume that $c$ belongs to $S$.
Clearly, $c$ can receive $4$.
We note that the black vertices of $S$ can still receive $3r_2 + r_3$;
and since $r_2 + 2r_3 \geq |\mathcal B|$, they can receive at least $\frac{1}{2}|\mathcal B| + \frac{5}{2}r_2$
(thus, at least $\frac{1}{2}|\mathcal B|$).

If $c'$ does not belong to $S$,
then we consider the graph induced by $c'$ and by the black and blue vertices,
and we observe that $r_2 + d \geq |\mathcal B| - 1$
(since there are at least $|\mathcal B| - 1$ components of $\mathcal B$ not containing $c'$).
Consequently, we have $\frac{1}{2}|\mathcal B| + \frac{5}{2}r_2 + 4 > |\mathcal B| + 5 - 4d$ since $|\mathcal B| \geq 2$, a contradiction.

We assume that $c'$ belongs to $S$,
so $c'$ can receive $4$.
If $d \geq 1$, then
the remaining charge is at most $|\mathcal B| + 1$;
and we have $\frac{1}{2}|\mathcal B| + 4 + 4 > |\mathcal B| + 1$ since $|\mathcal B| \leq 13$, a contradiction.

We assume that $d = 0$.
If there is not a component containing $x$ as its only vertex,
then the remaining charge is $|\mathcal B|$.
Similarly as above, we have $\frac{1}{2}|\mathcal B| + 4 + 4 > |\mathcal B|$, a contradiction.

We assume that $F^+_{1, 0} - S$ has a component consisting of $x$,
so all neighbours of $x$ belong to $S$.
Since $d = 0$ and since $|\mathcal B| \geq 2$, we have $r_2 \geq 1$.
If $|\mathcal B| \leq 11$, then we have $\frac{1}{2}|\mathcal B| + \frac{5}{2}r_2 + 4 + 4 \geq |\mathcal B| + 5$, a contradiction.
If $|\mathcal B| = 12$, then the parity implies that $r_2 \geq 2$ or $r_3 \geq 6$, and a contradiction follows.

We assume that $|\mathcal B| = 13$.
We consider the structure of $\mathcal B$ and we observe that $r_3 \leq 4$, that is, $r_2 \geq 5$.
Consequently, we get $\frac{1}{2}|\mathcal B| + \frac{5}{2}r_2 + 4 + 4 > |\mathcal B| + 5$,
and we obtain the desired distribution of the assigned charge, a contradiction.
Thus, $F^+_{1, 0}$ is $\frac{5}{4}$-tough.
\end{proof}

We continue with the following.

\begin{prop}\label{87tough}
The graphs $F^+_{2, 0}$ and $F_{2, 0}$ are $\frac{8}{7}$-tough.
\end{prop}
\begin{proof}
By Proposition~\ref{simpl}, it suffices to show
the toughness of $F^+_{2, 0}$ which we do via contradiction.
We suppose that
there is a separating set $S$ of vertices in $F^+_{2, 0}$
such that $c(F^+_{2, 0} - S) > \frac{7}{8}|S|$.

Since $S$ is separating, we have $|S| \geq 3$.
Consequently, we can assume that $c(F^+_{2, 0} - S) \geq 3$.

To specify the structure of $S$, we consider the graph $F_{2, 0}$.
We note that $F_{2, 0}$
contains two $T$-regions and the $T$-regions share their outer vertices;
and we call these outer vertices \emph{black}
(as well as the corresponding vertices of $F^+_{2, 0}$).
We observe that every pair of black vertices is dominating $F_{2, 0}$.

Since $c(F^+_{2, 0} - S) \geq 3$,
we have that at least two black vertices belong to $S$.
We note that $F^+_{2, 0}$ contains one $T$-region
(and its outer vertices are black),
and we modify $S$ using Proposition~\ref{Ttough};
and we let $S'$ be the resulting set.
Considering the possibilities,
we observe that $c(F^+_{2, 0} - S') \leq \frac{7}{8}|S'|$, a contradiction.
\end{proof}

We shall use the toughness of the building blocks $F_{i, 0}$
(given by Propositions~\ref{54tough} and~\ref{87tough} and by Corollary~\ref{32tough})
to show the toughness of the constructed graphs $F_{i, n}$.

\section{Gluing tough graphs}
\label{sg}

We shall use the following lemma as the main tool for showing the toughness of graphs
which are obtained by the standard construction for bounding the shortness exponent.

\begin{lemma}\label{constr2}
For $i = 1,2$, let $G^+_i$ and $G_i$ be $t$-tough graphs
such that $G_i$ is obtained by removing vertex $v_i$ from $G^+_i$.
Let $U$ be a graph obtained from the disjoint union of $G_1$ and $G_2$ by adding new edges
such that the minimum degree of the bipartite graph $(N(v_1), N(v_2))$ is at least $t$.
Then $U$ is $t$-tough.
\end{lemma}
\begin{proof}
We assume that $t > 0$ and that there exists a separating set of vertices in $U$.
We let $X$ be such a set and we let $X_i = X \cap V(G_i)$ for $i = 1,2$.
Clearly, $2 \leq c(U-X) \leq c(G_1-X_1) + c(G_2-X_2)$,
and we use it to show that $c(U-X) \leq \frac{1}{t}|X|$.

If $X_i$ is a separating set in $G_i$, then the toughness of $G_i$ implies that
$c(G_i-X_i) \leq \frac{1}{t}|X_i|$.

We suppose that $X_1$ is not separating in $G_1$,
and we observe that $c(U- X) \leq c(G_2-X_2) +1$.
If $c(U- X) \leq c(G_2-X_2)$, then $X_2$ is separating in $G_2$,
and thus, $c(G_2-X_2) \leq \frac{1}{t}|X_2|$
and the desired inequality follows since $|X_2| \leq |X|$.

We assume that $c(U- X) = c(G_2-X_2) +1$.
Clearly, if $N(v_2) \subseteq X_2$, then $c(G_2-X_2) +1 = c(G^+_2-X_2)$;
so $X_2$ is separating in $G^+_2$ and
we have $c(G^+_2-X_2) \leq \frac{1}{t}|X_2|$ and the inequality follows.

In addition, we assume that there is a vertex of $N(v_2)$ not belonging to $X_2$.
We recall that this vertex has at least $\lceil t \rceil$ neighbours in $N(v_1)$.
Since $c(U- X) = c(G_2-X_2) +1$, we note that all these neighbours belong to $X_1$.
Thus, $|X_1| \geq t$ and we have $c(G_1-X_1) \leq \frac{1}{t}|X_1|$.

Similarly, we get that if $X_2$ is not separating in $G_2$,
then $c(G_2-X_2) \leq \frac{1}{t}|X_2|$.
We conclude that (no matter whether $X_i$ is separating or not)
we have $c(G_i-X_i) \leq \frac{1}{t}|X_i|$ for both $i = 1,2$;
and the inequality follows.
\end{proof}

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.6]{counter.pdf}
    \caption{A counterexample to a statement presented in~\cite[Lemma~1]{5/4}.
    We consider the graphs $G^+_i$ and $G_i = G^+_i - v_i$ for $i = 1,2$, and the graph $U$.
    We note that $U$ is obtained from the disjoint union of $G_1$ and $G_2$
    by adding edges and every vertex of $N(v_1) \cup N(v_2)$ is incident with at least one new edge.
    Clearly, $G^+_i$ and $G_i$ are $\frac{3}{2}$-tough,
    but $U$ is not.}
    \label{f:counter}
\end{figure}

We remark that a similar statement appears in~\cite[Lemma~1]{5/4};
and it is used in~\cite{chpl}.
(Compared to Lemma~\ref{constr2},
the main difference is that the minimum degree
of the considered bipartite graph is required to be at least $1$.)
The graphs depicted in Figure~\ref{f:counter} show that this statement is false.
We view Lemma~\ref{constr2} as a fixed version of this statement,
and we remark that Lemma~\ref{constr2} can be applied in the arguments of~\cite{5/4} and~\cite{chpl}.
We note that Lemma~\ref{constr2} can be viewed as a generalization of a similar statement
(for $1$-tough graphs) presented in~\cite[Lemma~4]{dill}.

\section{Toughness of the constructed graphs}
\label{tt}

In this section, we clarify that the graphs $F_{i, n}$ have the properties
stated in Proposition~\ref{shortAll}.
\begin{proof}[Proof of Proposition~\ref{shortAll}]
We recall that the order of the constructed graphs and the length of their longest cycles are given
by Corollary~\ref{cyc123} and by Proposition~\ref{cyc4}.

For every $i = 1,2$, we show the toughness of the graphs $F_{i, n}$ using induction on $n$.
The case $n = 0$ is verified by Propositions~\ref{54tough} and~\ref{87tough}.
By induction hypothesis, $F_{i, n-1}$ has the required toughness,
and by Proposition~\ref{simpl}, so does a graph obtained from $F_{i, n-1}$ by removing a simplicial vertex;
we shall apply Lemma~\ref{constr2}, and we view these two graphs as playing the role of $G^+_1$ and $G_1$ and
we view graphs $F^+_{i, 0}$ and $F_{i, 0}$ as playing the role of $G^+_2$ and $G_2$.
We consider the graph obtained from $F_{i, n-1}$ by replacing one
of its simplicial vertices by a copy of $F_{i, 0}$ and by adding edges as in the present construction
(we recall the construction of graphs $F_{i, n}$ for $i = 1,2$; see Section~\ref{sc}).
By Lemma~\ref{constr2}, the resulting graph has the required toughness.
Thus, we can replace a simplicial vertex
of the resulting graph and apply Lemma~\ref{constr2} again;
and repeating this argument, we obtain that $F_{i, n}$ has the required toughness.

Similarly, we show the toughness of $F_{3, n}$ by induction on $n$.
By Corollary~\ref{32tough}, the toughness of $F_{3, 0}$ is greater than $1$.
We consider Lemma~\ref{constr3} (see below)
applied on the graph $F_{3, n-1}$ playing the role of $G$
(and then applied repeatedly on the resulting graph),
and we obtain that the toughness of $F_{3, n}$ is greater than $1$.
\end{proof}

Similarly to Lemma~\ref{constr2}, the following lemma can be used to
construct large graphs from smaller ones while preserving certain toughness.

\begin{lemma}\label{constr3}
Let $G$ be a graph of toughness greater than $1$ which contains a $K_4$-region
and let $G'$ be a graph obtained from $G$
by replacing this $K_4$-region by a $T$-region (in the natural way).
Then the toughness of $G'$ is greater than $1$.
\end{lemma}
\begin{proof}
We let $X'$ be a separating set of vertices in $G'$, and we shall show that $c(G'-X') < |X'|$.
We consider the set $X$ obtained from $X'$ by removing all inner vertices of the new $T$-region.
For the sake of simplicity,
we let $c = c(G' - X') - c(G - X)$ and $x = |X' \setminus X|$,
and we note that it suffices to show that $c(G-X)+c < |X|+x$.
Considering the choice of $X$, we observe that $c \leq x$.
We conclude the proof by showing that $c(G-X) < |X|$.
If $X$ is separating in $G$, then the inequality is given by the toughness of $G$.
We can assume that $c(G-X) = 1$. Consequently, $c(G'-X') > c(G-X)$,
so at least two outer vertices of the new $T$-region belong to $X'$.
Thus, they belong to $X$, that is, $|X| \geq 2$ and the inequality follows.
\end{proof}

\section{Note on longest paths}
\label{lp}

We remark that (using similar arguments as in Section~\ref{sc} we obtain that)
a longest path of $F_{i,n}$ has $p_i(n)$ vertices where
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item $p_1(n) = 2 + c_1(n) + 2\sum_{k=0}^{n-1} c_1(k)$,
\item $p_2(n) = 1 + \text{sgn}(n) + c_2(n) + c_2(n-1) + 2\sum_{k=0}^{n-2} c_2(k)$,
\item $p_3(n) = 7 \cdot 2^{n+2} + 2\text{sgn}(n) -15n -19$.
\end{enumerate}

\section*{Acknowledgements}

The author would like to thank Petr Vr\'{a}na for discussing different
strategies for the graph constructions,
and to thank the anonymous referees for their helpful comments.

The research was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports
and by the project 17-04611S of the Czech Science Foundation.


The Windows 10 has provided us with numerous applications which can easily be accessed right from the Start menu. One of the additions to the journey of Windows is the Photos app. Microsoft had launched it along with other additional features like the Windows Store in the Windows 8 version. The Windows 10 had emerged as a blend of its 7 and 8 versions, which is apparent from the Start menu at the desktop, clicking on which you get a tile of applications. Here, we will see some keyboard shortcuts for the Windows photos app. For all those out there who prefer their typing pads over the mouse pads for computation, trust me, this article will be of a great help to you. Once you get used to these, you are never going to leave it behind.
Once you click the Windows icon that appears at the bottom of your desktop, you can find the Photos app under Life at a Glance. By default, you find your recent photos getting displayed as a slide show over there. Clicking over there is going to take you to the collection of all pictures stored on your system. Read on to explore more!
Before starting, let me introduce you to the app. When you click on the 3 horizontal lines at the top, you will find three options Collections, Albums and Folders. Collections contain all the pictures in your system sorted by date, albums bring together those photos which are taken at a particular date and folders sort them according to My Pictures and One Drive.
Spacebar – Suppose that you are currently in your collections mode. Pressing this will help select pictures.
A checkbox appears on top of every photo as shown above. Check on them to select the pictures.
Enter Key– This will open up a photo from the collection to its viewing mode.
Ctrl Key + C- This helps you copy a photo.
Ctrl Key + O – This will help resetting photo to its original size after zooming it.
Ctrl Key + P – This is used to print the photo currently selected.
Ctrl Key + "+" – This is used to zoom a photo to a larger size when you are viewing it.
Ctrl Key + "-" – This is helpful in zooming out a photo to a smaller size.
Spacebar while viewing photos – This opens up the menu at top. The menu has options to Edit, Enhance , share the photo and much more.
Ctrl Key + S – If you have edited the photo, it helps you save it after modifications.
Ctrl Key + Y – If you had gone backward in editing the photos, it helps you redo the changes.
Ctrl Key + Z – This helps you undo changes after editing it.
Ctrl Key + L – If you have to set the photo you are currently viewing as the lock screen, then this helps.
Windows Key + H – This will help you open up the Share charm at the top. You can share your photo you are viewing via twitter or email.
E – This is used to enhance the photo, in terms of brightness.
Alt + Enter – This will open up the File info from the left, which has information like size, dimensions, device from which it was taken, date, etc.
F5 – This will start a slide show of all pictures.
Esc – It lets to move on to the previous screen.
Ctrl Key + R – This will rotate the picture in all four angles at each press of the keys.
That's it. Try out all the shortcuts mentioned above and have fun!.


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redpajama_set_name: RedPajamaCommonCrawl


Competitors participate in this Brisbane Eisteddfod according to all rules and guidelines listed and any other rules and conditions specific to Events as listed in the Individual Schedules.
The Event Management Team (EMT) reserves the right to allow variances of all rules and guidelines at their discretion. The EMT reserves the right to decide on any matter not covered by all listed rules and guidelines. Decisions of this nature will be deemed final without further question.
Volunteers and Adjudicator/s are not permitted to advise, interpret or vary any rules or guidelines. The EMT will not accept responsibility for any outcome arising from such unauthorized actions.
Entries must be received by Brisbane Eisteddfod Administration prior to the close date of each event. Entries must be completed correctly, comply with all rules listed in associated event schedules and be accompanied by full fees including program cost or they will not be processed. Responsibility for correct entry rests with the entrant. The Eisteddfod cannot be responsible for entries lost through failed electronic transmission. An entry acknowledgement receipt will be sent to the email address provided on the entry form, confirming the events entered and fees paid. Entrants must contact the Eisteddfod if this acknowledgement is not received within 3 days of lodgement.Entrants are advised to check the details carefully and notify the Eisteddfod immediately of any errors, including spelling of names, etc. Corrections cannot be made after final timetabling.
All Eisteddfod correspondence is via email. It is the entrant's responsibility to ensure that Brisbane Eisteddfod emails are not blocked by spam filters.
Entry Fees and Program costs are not refundable except when an event is cancelled. Entry fees and program costs cannot be transferred to another event should entrants change their mind after final timetabling for the initial entry.
Entries submitted after the commencement date of the Eisteddfod will be subject to an additional (late) fee of $25. Late entries are only accepted without obligation, subject to final timetabling and capacity.
Purchase of an Event Program is compulsory and is added to the invoice at the time of entry. Teachers submitting entries for individual students must include the cost of an event program for each student, except where students share the same family address. Event Programs will be mailed 14 prior days to the event commencement date. In the case of non-delivery of an event program the entrant will be provided with an equitable replacement.
Please email admin@brisbaneeisteddfod.org.au if the program has not arrived within seven days of Event start date.
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In the interest of fairness no competitor will be allowed to perform if they have had professional contact with any adjudicator/s in the six months prior to the start date of the Eisteddfod.
Personal communication with or other representation to the adjudicator/s by competitors, teachers, parents, friends or associated parties is strictly prohibited and may result in disqualification at the Committee's discretion.
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All mobile phones/devices with screens must be turned off within the Auditorium.
A protest of any nature must be made in writing and lodged with the EMT within one hour of the conclusion of the section in which the disputed matter arises. Protests will only be accepted from teachers and attract a $25 fee. The fee is refundable if the protest is upheld. Any protest will be referred to a Protest EMT for settlement and the resulting decision will be final and accepted without further question.
The EMT will not be liable for personal items lost or stolen during the course of the Eisteddfod. Lost property will be held for a period of one month after the final day of the Eisteddfod.
TICKET SALES: Tickets are non-refundable. All patrons (including children 3 years & over) must hold a valid ticket.
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