An Upper Bound for the Regularity of Symbolic Powers of Edge Ideals of Chordal Graphs

Seyed Amin Seyed Fakhari

doi:10.37236/8566

Abstract

Assume that $G$ is a chordal graph with edge ideal $I(G)$ and ordered matching number $\nu_{o}(G)$. For every integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})\leq 2s+\nu_{o}(G)-1$. As a consequence, we determine the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs.

Highlights

Let K be a field and S = K[x1, . . . , xn] be the polynomial ring in n variables over K
There is a natural correspondence between quadratic squarefree monomial ideals of S and finite simple graphs with n vertices
Gu, Ha, O’Rourke and Skelton [9] proved the same inequality for symbolic powers

Summary

Introduction

Gu, Ha, O’Rourke and Skelton [9] proved the same inequality for symbolic powers More explicit, they proved that reg(I(G)(s)) 2s + ind-match(G) − 1, for any graph G and any integer s 1. Haand Van Tuyl [11, Theorem 6.7] proved that for every graph G, reg(I(G)) match(G) + 1, where match(G) denotes the matching number of G. This inequality was strengthen by Constantinescu and Varbaro [7, Remark 4.8] (see [19, Corollary 2.5]). We determine the regularity of symbolic powers of edge ideals of graphs belonging to the following classes.

Preliminaries

Main results