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

Marcel Morales, Nguyen Thi Dung

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 $I^{k}_G$. 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$(I_G)$.

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

Abstract

Let be a simple, connected non bipartite graph and let be the edge idealof . 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 . 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 in terms of some subsets of . We give explicit formulas for the numbers astab and dstab, which are, respectively, the smallest number such that for all and the smallest number such that the maximal ideal belongs to . We also give very simple upper bounds for astab and dstab.

Paper Structure

This paper contains 5 sections, 23 theorems, 37 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing Factor-critical graphs by replication
  4. Stable associated primes, dstab and astab
  5. $G$ has an optimal ear decomposition starting with an odd cycle.

Key Result

Theorem 1

Let $G$ be a simple, connected non bipartite graph. Then $\mathfrak m ^{{\bf 1}_{U}} \in \mathrm{Ass} (I^{\mathrm{astab}(I_G)}_G)$ if and only if $Z:= V\setminus U$ is a coclique set and either $U=N(Z)$ or $U\not=N(Z)$ and every connected component of $G_{ U\setminus N(Z)}$ contains an odd cycle.

Figures (11)

  • Figure 1: $G$ and a generalized ear decomposition
  • Figure 2: $G/F$ contraction of the edges $F=\{cd,dg,eh\}$
  • Figure 3: $G\succ F$ Subdivision of the edges $F=\{cd,dg,eh\}$
  • Figure 4: $G$ and $p_{{\bf a}+ {\bf 1}_G}(G)$
  • Figure 5: Identification of a duplicated vertex
  • ...and 6 more figures

Theorems & Definitions (55)

  • Theorem
  • Definition
  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • ...and 45 more