Normality and associated primes of Closed neighborhood ideals and dominating ideals

TL;DR

This paper investigates the normality and torsion properties of specific monomial ideals associated with graph structures, providing criteria and demonstrating these properties for various classes of graphs.

Contribution

It introduces new sufficient criteria for normality of monomial ideals and applies them to closed neighborhood and dominating ideals of certain graphs.

Findings

Closed neighborhood ideals of complete bipartite graphs are normal.

Dominating ideals of complete bipartite graphs are nearly normally torsion-free.

Under certain conditions, dominating ideals of h-wheel graphs are normal.

Abstract

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of $h$-wheel graphs, under certain condition, are normal.