Joins, Ears and Castelnuovo-Mumford regularity

Abstract

We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[E G ] be the polynomial ring on the edges of G and K[V G ] the polynomial ring on the vertices of G. We associate to G an ideal, I(X G ), defined as the preimage of (xwhich sends a variable, te, associated to an edge e = {i, j}, to the product x i x j of the variables associated to its vertices. We show that K[E G ]/I(X G ) is a one-dimensional, Cohen-Macaulay, graded ring, that I(X G ) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo-Mumford regularity of I(X G ) providing the following sharp upper and lower bounds:where µ(G) is the maximum vertex join number of the graph and b 0 (G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.