Ordered alternating paths and the depth of symbolic powers of cover ideals of graphs

Abstract

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell (M)$ the length of the longest $M$-alternating path in $G$. We define ${\alpha }_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell (M)\le 2t-1$. We then prove that $$\mathrm{depth}(S/J(G{)}^{(t)})\le |V(G)|-1-{\alpha }_t(G)$$ for all $t\ge 1$, where $J(G{)}^{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.