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We introduce a conjecture about constructing critically (s+1)-chromaticgraphs from critically s-chromatic graphs. We then show how this conjectureimplies that any unmixed height two square-free monomial ideal I, i.e., thecover 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), thefractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <=\chi(G). We give an algebraic proof of this result.

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