We introduce a conjecture about constructing critically -chromatic graphs from critically -chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal in a polynomial ring , i.e., the cover ideal of a finite simple graph, has the persistence property, that is, for all . To support our conjecture, we prove that the statement is true if we also assume that , the fractional chromatic number of the graph , satisfies . We give an algebraic proof of this result.