Herzog, Hibi and Ohsugi conjecture for trees

TL;DR

This paper proves Herzog, Hibi and Ohsugi's conjecture for trees, showing all powers of vertex cover ideals of trees are componentwise linear, and extends results to certain unicyclic graphs.

Contribution

It establishes the conjecture for trees and identifies conditions under which symbolic powers of vertex cover ideals are componentwise linear in unicyclic graphs.

Findings

All powers of vertex cover ideals of trees are componentwise linear.

Symbolic powers of vertex cover ideals are componentwise linear for certain unicyclic graphs.

The conjecture holds for a broader class of graphs under specific conditions.

Abstract

Let $S=\mathbb{K}[x_1,\dots, x_n]$ be a polynomial ring, where $\mathbb{K}$ is a field, and $G$ be a simple graph on $n$ vertices. Let $J(G)\subset S$ be the vertex cover ideal of $G$. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if $G$ is a unicyclic vertex decomposable graph that does not contain $C_3$ or $C_5$, then symbolic powers of $J(G)$ are componentwise linear.