The v-numbers and linear presentations of ideals of covers of graphs

Abstract

Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of this work are to study the {\rm v}-number $\mathrm{v}(J)$ of $J$ and to study when $J$ is linearly presented using combinatorics and commutative algebra. We classify when $\mathrm{v}(J)$ attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If $G$ is unmixed, the graph ${\mathcal{G}}_J$ of $J$ is the graph whose vertices are the minimal vertex covers of $G$ and whose edges are the pairs $\{C,C^{\prime }\}$ such that $|C\cup C^{\prime }|=|C|+1$. We show necessary and sufficient conditions for the graph ${\mathcal{G}}_J$ of $J$ to be connected. Then, for unmixed König graphs, we classify when $J$ is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If $G$ is unmixed, it is shown that the columns of the linear syzygy matrix of $J$ are linearly independent if and only if ${\mathcal{G}}_J$ has no strong $3$-cycles. One of our main theorems shows that if $G$ is unmixed and has no induced $4$-cycles, then $J$ is linearly presented. For unmixed graphs without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly presented.

Bull. Malays. Math. Sci. Soc., to appear