  Let $G$ be an (oriented) graph on the vertex set $V = \{ 0, 1,\ldots,n\}$
with root $0$. Postnikov and Shapiro associated a monomial ideal
$\mathcal{M}_G$ in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over
a field $\mathbb{K}$. A subideal $\mathcal{M}_G^{(k)}$ of $\mathcal{M}_G$
generated by subsets of $\widetilde{V}=V\setminus \{0\}$ of size at most $k+1$
is called a $k$-skeleton ideal of the graph $G$. Many interesting homological
and combinatorial properties of $1$-skeleton ideal $\mathcal{M}_G^{(1)}$ are
obtained by Dochtermann for certain classes of simple graph $G$. A finite
sequence $\mathcal{P}=(p_1,\ldots,p_n) \in \mathbb{N}^n$ is called a spherical
$G$-parking function if the monomial $\mathbf{x}^{\mathcal{P}} =
\prod_{i=1}^{n} x_i^{p_i} \in \mathcal{M}_G \setminus \mathcal{M}_G^{(n-2)}$.
Let ${\rm sPF}(G)$ be the set of all spherical $G$-parking functions. In this
paper, a combinatorial description for all multigraded Betti numbers of the
$k$-skeleton ideal $\mathcal{M}_{K_{n+1}}^{(k)}$ of the complete graph
$K_{n+1}$ on $V$ are given. Also, using DFS burning algorithms of
Perkinson-Yang-Yu (for simple graph) and Gaydarov-Hopkins (for multigraph), we
give a combinatorial interpretation of spherical $G$-parking functions for the
graph $G = K_{n+1}- \{e\}$ obtained from the complete graph $K_{n+1}$ on
deleting an edge $e$. In particular, we showed that $|{\rm sPF}(K_{n+1}-
\{e_0\} )|= (n-1)^{n-1}$ for an edge $e_0$ through the root $0$, but $|{\rm
sPF}(K_{n+1} - \{e_1\})| = (n-1)^{n-3}(n-2)^2$ for an edge $e_1$ not through
the root.

Combinatorics;Commutative Algebra