A conjecture on critical graphs and connections to the persistence of associated primes
Abstract
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/Is) ⊂eq Ass(R/Is+1) for all s >= 1. To support our conjecture, we prove that the statement is true if we also assume that f(G), the fractional chromatic number of the graph G, satisfies (G) -1 < f(G) <= (G). We give an algebraic proof of this result.
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