Unmixed Graphs that are Domains

TL;DR

This paper characterizes graphs whose basic covers algebra is a domain, extending previous bipartite results to more general graphs, and relates this to properties of their edge ideals and symbolic powers.

Contribution

It provides a combinatorial characterization of graphs with domain basic covers algebra and unmixed edge ideals, generalizing prior bipartite-specific results.

Findings

Characterization of graphs with domain basic covers algebra

Conditions when the edge ideal is unmixed and the algebra is a domain

Complete description of graphs with all symbolic powers generated in the same degree

Abstract

Given an arbitrary graph G, we study its basic covers algebra, which is the symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the case when G is bipartite, we characterize in a combinatorial fashion the situations when: 1) the basic covers algebra is a domain, and 2) it is a domain and in addition (the edge ideal of) G is unmixed. It turns out that the last result gives a complete characterization of those graphs for which any symbolic power of the edge ideal is generated by monomials of the same degree.