A conjecture on critical graphs and connections to the persistence of associated primes

作者：Francisco Christopher A; Ha Huy Tai; Van Tuyl Adam^*

来源：Discrete Mathematics, 2010, 310(15-16): 2176-2182.

DOI：10.1016/j.disc.2010.04.014

• Associated primes
• Monomial ideals

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)) subset of 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 chi(f) (G), the fractional chromatic number of the graph G, satisfies chi(G) - 1 < chi(f)(G) <= x (G). We give an algebraic proof of this result.

• 出版日期2010-8-28