Replication in critical graphs and the persistence of 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.
Keywords
Cite
@article{arxiv.1301.6983,
title = {Replication in critical graphs and the persistence of monomial ideals},
author = {Tomáš Kaiser and Matěj Stehlík and Riste Škrekovski},
journal= {arXiv preprint arXiv:1301.6983},
year = {2014}
}