RESOLUTION OF UNMIXED BIPARTITE GRAPHS
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Let G be a graph on the vertex set V (G) = {x1, . . . , xn} with the edge set E(G), and let R = K[x1, . . . , xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials xixj with {xi, xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈C xi 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 LG and we explicitly describe the minimal free resolution of the ideal associated to LG 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.
Abstract
Let G be a graph on the vertex set V (G) = {x1, . . . , xn} with the edge set E(G), and let R = K[x1, . . . , xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials xixj with {xi, xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈C xi 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 LG and we explicitly describe the minimal free resolution of the ideal associated to LG 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.
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