On the regularity of edge ideal of graphs

TL;DR

This paper investigates the regularity of edge ideals of graphs, introducing new invariants and establishing bounds and equalities for specific classes of graphs, including Cohen–Macaulay and doubly Cohen–Macaulay graphs.

Contribution

It introduces the invariants $ ext{ind-match}_{ ext{H}}(G)$ and $ ext{min-match}_{ ext{H}}(G)$, and proves bounds and exact formulas for the regularity of edge ideals in various graph classes.

Findings

Bounds the regularity between $ ext{ind-match}_{ ext{H}}(G)$ and $ ext{min-match}_{ ext{H}}(G)$.

Shows equality of regularity and invariants for Cohen–Macaulay graphs with girth at least five.

Establishes conditions for equality in doubly Cohen–Macaulay graphs.

Abstract

Let $G$ be a graph with $n$ vertices, $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$ and $I(G)$ denote the edge ideal of $G$. For every collection $\mathcal{H}$ of connected graphs with $K_2\in \mathcal{H}$, we introduce the notions of $\ind-match_{\mathcal{H}}(G)$ and $\min-match_{\mathcal{H}}(G)$. It will be proved that the inequalities $\ind-match_{\{K_2, C_5\}}(G)\leq{\rm reg}(S/I(G))\leq\min-match_{\{K_2, C_5\}}(G)$ are true. Moreover, we show that if $G$ is a Cohen--Macaulay graph with girth at least five, then ${\rm reg}(S/I(G))=\ind-match_{\{K_2, C_5\}}(G)$. Furthermore, we prove that if $G$ is a paw--free and doubly Cohen--Macaulay graph, then ${\rm reg}(S/I(G))=\ind-match_{\{K_2, C_5\}}(G)$ if and only if every connected component of $G$ is either a complete graph or a $5$-cycle graph. Among other results, we show that for every doubly Cohen--Macaulay simplicial complex, the equality ${\rm reg}(\mathbb{K}[\Delta])={\rm dim}(\mathbb{K}[\Delta])$ holds.