Matchings, coverings, and Castelnuovo-Mumford regularity

TL;DR

This paper establishes a connection between graph covering parameters and the Castelnuovo-Mumford regularity of edge ideals, providing new and existing bounds through combinatorial graph theory insights.

Contribution

It introduces a novel upper bound for regularity based on the co-chordal cover number and derives new bounds using additional graph covering results.

Findings

Co-chordal cover number bounds regularity from above.

Several known bounds are simplified as consequences of graph coverings.

New upper bounds for regularity are established using advanced covering results.

Abstract

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.