Christopher A. Francisco
Associated primes of monomial ideals and odd holes in graphs.
Mon, 01/11/2010 - 16:01 — SomNambulAuthors: Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl
Subjects: Commutative Algebra
AbstractLet $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote
the Alexander dual of $I(G)$. We show that a description of all induced cycles
of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This
result forms the basis for a method to detect odd induced cycles of a graph via
ideal operations, e.g., intersections, products and colon operations. Moreover,
we get a simple algebraic criterion for determining whether a graph is perfect.
We also show how to determine the existence of odd holes in a graph from the
value of the arithmetic degree of $J(G)^2$.A conjecture on critical graphs and connections to the persistence of associated primes.
Wed, 11/11/2009 - 16:01 — SomNambulAuthors: Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl
Subjects: Commutative Algebra
AbstractWe 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/I^s) \subseteq 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) <=
\chi(G).
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