Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

TL;DR

This paper explores the algebraic and combinatorial properties of skeleton ideals associated with graphs, especially complete graphs and their subgraphs, introducing new descriptions of Betti numbers and parking functions.

Contribution

It provides a combinatorial description of Betti numbers for skeleton ideals of complete graphs and interprets spherical parking functions for graphs with deleted edges.

Findings

Betti numbers of skeleton ideals are explicitly described for complete graphs.

Spherical parking functions are characterized for graphs obtained by removing an edge from a complete graph.

The number of spherical parking functions varies depending on whether the removed edge is incident to the root.

Abstract

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.