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 $I$ in a polynomial ring $R$, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, $\mathrm{Ass}(R/I^s)\subseteq \mathrm{Ass}(R/I^{s+1})$ for all $s\ge 1$. To support our conjecture, we prove that the statement is true if we also assume that ${\chi }_f\left(G\right)$, the fractional chromatic number of the graph $G$, satisfies $\chi \left(G\right)-1<{\chi }_f\left(G\right)\le \chi \left(G\right)$. We give an algebraic proof of this result.