Regularity of the vanishing ideal over a bipartite nested ear decomposition

Abstract

We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length a increases by a Sa 2a (q - 2), where q is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, G, endowed with a weak nested ear decomposition is equal to |VG| + - 3 2 (q - 2), where is the number of even length ears and pendant edges of the decomposition. As a corollary, we show that for bipartite graph the number of even length ears in a nested ear decomposition starting from a vertex is constant.