Improved bounds for the regularity of powers of edge ideals of graphs

TL;DR

This paper establishes new upper and lower bounds for the regularity of powers of edge ideals of graphs, using graph invariants like matching numbers, and constructs graphs that demonstrate strict inequalities among these bounds.

Contribution

It introduces improved bounds for the regularity of powers of edge ideals based on graph invariants and constructs graphs that exhibit strict inequalities among these bounds.

Findings

New upper bounds for ${ m reg}(I(G)^s)$ involving $ ext{min-match}$ and $ ext{ord-match}$.

Lower bounds for ${ m reg}(I(G)^s)$ using $ ext{ind-match}_{ ext{K}_2, ext{C}_5}$.

Existence of graphs with strict inequalities among bounds, answering an open question.

Abstract

Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of $\min-match_{\{K_2, C_5\}}(G)$ and $\ind-match_{\{K_2, C_5\}}(G)$ from \cite{sy}. We show that $${\rm reg}(I(G)^s)\leq 2s+\min-match_{\{K_2, C_5\}}(G)-1,$$for all $s\geq 1$, which implies that$${\rm reg}(I(G)^s)\leq 2s+\min-match(G)-1.$$Moreover, we show that$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-2,$$and if $\ind-match_{\{K_2, C_5\}}(G)$ is an odd integer, then$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-1.$$Furthermore, it is shown that$${\rm reg}(I(G)^s)\leq 2s+\ord-match(G)-1,$$where $\ord-match(G)$ denotes the ordered matching number of $G$. Finally, we construct infinitely many connected graphs which satisfy the following strict inequalities:$$2s+\ind-match(G)-1 < {\rm reg}(I(G)^s)< 2s+{\rm cochord}(G)-1.$$This gives a positive answer to a question asked in \cite{jns}.