• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.977-986
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• 2015

Let G be a graph on the vertex set $V(G)=\{x_1,{\cdots},x_n\}$ with the edge set E(G), and let $R=K[x_1,{\cdots},x_n]$ 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 $\{x_i,x_j\}{\in}E(G)$, and the vertex cover ideal $I_G$ generated by monomials ${\prod}_{x_i{\in}C}{^{x_i}}$ for all minimal vertex covers C of G. A 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 $L_G$ 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.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.85-101
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• 2018

Let S be a polynomial ring over a field K, with a monomial order ${\prec}$, and let I be an unmixed graded ideal of S. In this paper we study two functions associated to I: The minimum distance function ${\delta}_I$ and the footprint function $fp_I$. It is shown that ${\delta}_I$ is positive and that $fp_I$ is positive if the initial ideal of I is unmixed. Then we show that if I is radical and its associated primes are generated by linear forms, then ${\delta}_I$ is strictly decreasing until it reaches the asymptotic value 1. If I is the edge ideal of a Cohen-Macaulay bipartite graph, we show that ${\delta}_I(d)=1$ for d greater than or equal to the regularity of S/I. For a graded ideal of dimension ${\geq}1$, whose initial ideal is a complete intersection, we give an exact sharp lower bound for the corresponding minimum distance function.