On the powers of vertex cover ideals

Abstract

Let S=K[x1,…,xn] be a polynomial ring, where K is a field, and G be a simple graph on n vertices. Let J(G)⊂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, then symbolic powers of J(G) are componentwise linear. © 2021