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

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. Lovász'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 $I_G^k$. 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 ${\cup }_{k=1}^{\infty }\mathrm{Ass}(I_G^k)$ 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 $\mathrm{Ass}(I_G^k)=\mathrm{Ass}(I_G^{k+i})$ for all $i\ge 0$ and the smallest number $k$ such that the maximal ideal belongs to $\mathrm{Ass}(I_G^k)$. We also give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$.