Bulletin of the
Korean Mathematical Society BKMS

Let $G$ be a graph on the vertex set $V(G)=\{x_1,\ldots, x_n\}$ with the edge set $E(G)$, and let $R=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$. Two monomial ideals are associated to $G$, the \textit{edge ideal} $I(G)$ generated by all monomials $x_i x_j$ with $\{x_i,x_j\}\in E(G)$, and the \textit{vertex cover ideal} $I_G$ generated by monomials $\prod_{x_i\in C} x_i$ for all minimal vertex covers $C$ of $G$. A {\em minimal vertex cover} of $G$ is a subset $C\subset V(G)$ 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 $\mathcal{L}_G$ and we explicitly describe the minimal free resolution of the ideal associated to $\mathcal{L}_G$ 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.

Keywords: minimal free resolution, unmixed bipartite graph, edge ideal

MSC numbers: Primary 13D02, 03G10; Secondary 05C25