A conjecture on critical graphs and connections to the persistence of associated primes

Abstract

a b s t r a c tWe introduce a conjecture about constructing critically (s + 1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I in a polynomial ring R, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I s ) ⊆ Ass(R/I s+1 ) for all s ≥ 1. To support our conjecture, we prove that the statement is true if we also assume that χ f (G), the fractional chromatic number of the graph G, satisfies χ (G) − 1 < χ f (G) ≤ χ (G). We give an algebraic proof of this result.

Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth

“…We choose to use the terminology of critically s-chromatic graphs to be consistent with [7], where Conjecture 1.1 was stated. Fig.…”

Powers of Monomial Ideals and Combinatorics

“…A vertex cover is minimal if no proper subset is also a vertex cover. Denote J = J(H) the cover ideal of H, which is generated by the square-free monomials corresponding to the minimal vertex covers of H. Francisco, Ha and Van Tuyl propose a conjecture related to the chromatic number of a graph G, and prove the persistence property of Ass(R/J(G) n ) provided that the conjecture holds (see [14,Theorem 2.6]). In another paper, they give an explicit description of all associated primes of Ass(R/J(H) n ), for any fixed number n ≥ 1, in terms of the coloring properties of hypergraphs arising from H, see [15,Corollary 4.5].…”

Monomial Ideals, Computations and Applications

“…The conjecture is true for all odd cycles (i.e., the critically 3-chromatic graphs; see [66] for a proof) and for graphs whose fractional chromatic number f .G/ (see Sect. The conjecture is true for all odd cycles (i.e., the critically 3-chromatic graphs; see [66] for a proof) and for graphs whose fractional chromatic number f .G/ (see Sect.…”