Resolution of unmixed bipartite graphs

Abstract

Let G be a graph on the vertex set V(G) = {x 1,…,x n } with the edge set E(G), and let R = K[x 1,…, x n ] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials x i x j with {x i,x j } ∈ E(G), and the vertex cover ideal I G generated by monomials ∏ x i ∈C x i for all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂ V(G) such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers L G and we explicitly describe the minimal free resolution of the ideal associated to L G which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.