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 / 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.