Factor-critical graphs and dstab, astab for an edge ideal

TL;DR

This paper explores the properties of factor-critical graphs and their relation to the stability indices of edge ideals, providing explicit formulas and bounds for these algebraic invariants using graph theoretical tools.

Contribution

It introduces a new method to make a graph factor-critical and derives explicit formulas and bounds for astab and dstab of edge ideals based on graph structure.

Findings

Explicit formulas for astab and dstab of edge ideals.

New graph-theoretic method for factor-critical graphs.

Simple upper bounds for stability indices.

Abstract

Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge idealof $G$. In our previous work we showed that L. Lov\'asz's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph given by Edmonds and Gallai are basic tools for the irreducible decomposition of $I^{k}_G$. In this paper we use some tools from graph theory, mainly Withney's theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set $\cup_ {k=1}^{\infty}{\rm Ass} (I^{k}_G)$ in terms of some subsets of $G$. We give explicit formulas for the numbers astab$(I_G)$ and dstab$(I_G)$, which are, respectively, the smallest number $k$ such that ${\rm Ass} (I^{k}_G)= {\rm Ass} (I^{k+i}_G)$ for all $i\geq 0$ and the smallest number $k$ such that the maximal ideal belongs to ${\rm Ass}(I^{k}_G)$. We also give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$.