Toric rings arising from vertex cover ideals

TL;DR

This paper extends the concept of sortability to a broader class of monomial ideals and constructs normal Cohen-Macaulay toric rings from vertex cover ideals of graphs, introducing a new graph operation called clique multi-whiskering.

Contribution

It generalizes sortability to non-uniform monomial ideals and introduces clique multi-whiskering, producing vertex cover ideals with desirable algebraic properties.

Findings

Normal Cohen-Macaulay toric rings are associated with vertex cover ideals.

Clique multi-whiskering always yields vertex cover ideals with componentwise linear powers.

Extension of sortability concept broadens understanding of monomial ideals.

Abstract

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.