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

TL;DR

This paper proposes a conjecture linking critical graph construction to the persistence of associated primes in monomial ideals, supported by algebraic proofs under specific fractional chromatic number conditions.

Contribution

It introduces a new conjecture connecting graph coloring properties with algebraic persistence, providing proofs under fractional chromatic number constraints.

Findings

Conjecture relates critical graph construction to algebraic persistence.

Proves the conjecture holds when fractional chromatic number satisfies certain bounds.

Establishes a link between graph theory and algebraic properties of monomial ideals.

Abstract

We 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, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I^s) \subseteq 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 \chi_f(G), the fractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <= \chi(G). We give an algebraic proof of this result.