Cohen-Macaulay binomial edge ideals and accessible graphs

TL;DR

This paper explores the relationship between Cohen-Macaulay binomial edge ideals and accessible graphs, establishing new combinatorial characterizations and conjecturing equivalence for a broad class of graphs.

Contribution

It introduces accessible graphs and proves that Cohen-Macaulay binomial edge ideals imply accessibility, providing a combinatorial framework for understanding Cohen-Macaulayness.

Findings

Cohen-Macaulay binomial edge ideals imply accessible graphs.

The conjecture that accessibility characterizes Cohen-Macaulayness is proven for chordal and traceable graphs.

A new combinatorial condition called strong unmixedness is shown to be equivalent to Cohen-Macaulayness.

Abstract

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen-Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.