Replication in critical graphs and the persistence of monomial ideals

TL;DR

This paper disproves a conjecture about the replication of vertices in critical graphs, providing counterexamples that also address questions about the depth and persistence of square-free monomial ideals.

Contribution

It introduces an infinite family of counterexamples to a conjecture on graph replication and resolves related questions on monomial ideal properties.

Findings

Counterexamples to the replication conjecture for critical graphs

The smallest counterexample addresses a question on depth functions

Results impact understanding of persistence in monomial ideals

Abstract

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose replication produces a (k+1)-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.) We disprove the conjecture by providing an infinite family of counterexamples. Furthermore, the smallest member of the family answers a question of Herzog and Hibi concerning the depth functions of square-free monomial ideals in polynomial rings, and a related question on the persistence property of such ideals.