On powers of the cover ideals of graphs

TL;DR

This paper investigates algebraic properties of powers of cover ideals in graphs, proving equalities for symbolic and ordinary powers in odd cycles, and exploring conditions for vertex decomposability and polymatroidality.

Contribution

It establishes the equality of symbolic and ordinary powers of cover ideals for odd cycles and links weak polymatroidality to vertex decomposability and clique-whiskered graphs.

Findings

$(J(C)^{(s)}) = (J(C)^s)$ for all $s \\geq 1$ and odd cycles

Vertex decomposability linked to weak polymatroidality of dual ideals

Powers of cover ideals in clique-whiskered graphs are weakly polymatroidal

Abstract

For a simple graph $G$, assume that $J(G)$ is the vertex cover ideal of $G$ and $J(G)^{(s)}$ is the $s$-th symbolic power of $J(G)$. We prove that $(J(C)^{(s)})=(J(C)^s)$ for all $s\geq 1$ and for all odd cycle $C$. For a simplicial complex $\Delta$, we show that $\Delta$ is vertex decomposable if $I_{\Delta}^{\vee}$ is weakly polymatroidal. Let $W=G^{\pi}$ be a fully clique-whiskering graph, we prove that $J(W)^s$ is weakly polymatroidal for all $s\geq 1$.