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, 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 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.

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