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

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 in a polynomial ring R, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/Is)⊆Ass(R/Is+1) for all s≥1. To support our conjecture, we prove that the statement is true if we also assume that χf(G), the fractional chromatic number of the graph G, satisfies χ(G)−1<χf(G)≤χ(G). We give an algebraic proof of this result.