Powers of Ideals Associated to $(C_4, 2K_2)$-free Graphs

TL;DR

This paper proves that for certain $(C_4, 2K_2)$-free graphs, all powers of their edge ideals have linear resolutions, and describes the regularity of powers of their vertex cover ideals in terms of graph degree.

Contribution

It establishes linear resolutions for all powers of edge ideals and linear quotients for all powers of vertex cover ideals in $(C_4, 2K_2)$-free graphs, providing a clear regularity description.

Findings

All powers of the edge ideal have linear resolutions for s ≥ 2.

Every power of the vertex cover ideal has linear quotients.

Regularity of powers of the vertex cover ideal depends on maximum degree of G.

Abstract

Let $G$ be a $(C_4, 2K_2)$-free graph with edge ideal $I(G)\subset \Bbbk[x_1,\dots , x_n]$. We show that $I(G)^s$ has linear resolution for every $s\geq 2$. Also, we show that every power of the vertex cover ideal of $G$ has linear quotients. As a result, we describe the Castelnuovo-Mumford regularity of powers of $I(G)^{\vee}$ in terms of the maximum degree of $G$.