Joins, ears and Castelnuovo–Mumford regularity

Abstract

We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[EG] be the polynomial ring on the edges of G and K[VG] the polynomial ring on the vertices of G. We associate to G an ideal, I(XG), defined as the preimage of (xi2−xj2:i,j∈VG)⊆K[VG] by the map K[EG]→K[VG] which sends a variable, te, associated to an edge e={i,j}, to the product xixj of the variables associated to its vertices. We show that K[EG]/I(XG) is a one-dimensional, Cohen–Macaulay, graded ring and we relate its degree with the number of vertices and connected components of the graph. We show that I(XG) 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(XG) providing the following sharp upper and lower bounds:μ(G)≤regI(XG)≤|VG|−b0(G)+1, where μ(G) is the maximum vertex join number of the graph and b0(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.