Improved bounds for the regularity of edge ideals of graphs
type: Item
instance of: scholarly article
title: Improved bounds for the regularity of edge ideals of graphs
zbMATH Open document ID: 1391.13024
author: S. A. Seyed Fakhari
author: Siamak Yassemi
published in: Collectanea Mathematica
publication date: 2018-05-24T00:00:00Z
review text: Let \(S=\mathbb{K}[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(\mathbb{K}\). Let \(G\) be a simple graph with vertex set \(V(G) = \{v_1,\dots,v_n\}\), edge set \(E(G)\) and the edge ideal \(I(G)\). A subset \(M\subseteq E(G)\) is a matching if \(e\cap e'=\emptyset\), for every pair of edges \(e,e'\in M\). The cardinality of the largest matching of \(G\) is called the matching number of \(G\) and is denoted by \(\mathrm{match}(G)\). The minimum cardinality of the maximal matchings of \(G\) is the minimum matching number of \(G\) and is denoted by \(\mathrm{min-match}(G)\). A matching \(M\) of \(G\) is an induced matching of \(G\) if for every pair of edges \(e,e'\in M\) there is no edge \(f\in E(G)\setminus M\) with \(f\subset e\cup e'\). The cardinality of the largest induced matching of \(G\) is called the induced matching number of \(G\) and is denoted by \(\mathrm{ind-match}(G)\). Let \(\mathcal{H}\) be a collection of connected graphs with \(K_2\in\mathcal{H}\). In this article under review, the authors introduce the notions of \(\mathrm{ind-match}_{\mathcal{H}}(G)\) and \(\mathrm{min-match}_{\mathcal H}(G)\) and they prove that the inequalities \(\mathrm{ind-match}_{\{K_2,C_5\}}(G)\leq \mathrm{reg}(S/I(G))\leq\mathrm{min-match}_{\{K_2,C_5\}}(G)\) are true. Moreover, the authors show that if \(G\) is a Cohen-Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))=\mathrm{ind-match}_{\{K_2,C_5\}}(G)\). Furthermore, they show that for every doubly Cohen-Macaulay simplicial complex, the equality \(\mathrm{reg}(\mathbb{K}[\Delta])=\dim(\mathbb{K}[\Delta])\) holds.
reviewed by: Amir Mafi
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 6872994
zbMATH Keywords: edge ideal
zbMATH Keywords: Castelnuovo-Mumford regularity
zbMATH Keywords: girth
zbMATH Keywords: matching
zbMATH Keywords: paw-free graph
MaRDI profile type: Publication
full work available at URL: s13348-017-0204-8
OpenAlex ID: W2754061169
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DOI: 10.1007/S13348-017-0204-8
description: scientific article; zbMATH DE number 6872994