Cohen-Macaulay binomial edge ideals of small graphs

TL;DR

This paper investigates the Cohen-Macaulay property of binomial edge ideals in small graphs, proving a conjecture for graphs up to 12 vertices and developing algorithms to check larger graphs, extending prior computational results.

Contribution

The paper proves a conjecture relating Cohen-Macaulayness to cut sets for graphs up to 12 vertices and creates algorithms to verify the conjecture for larger graphs, advancing computational methods.

Findings

Proved the conjecture for all graphs with up to 12 vertices.

Developed an algorithm to check the conjecture for graphs with up to 15 vertices.

Extended previous computational results significantly.

Abstract

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal $J_G$ to special disconnecting sets of vertices of its underlying graph $G$, called \textit{cut sets}. More precisely, the conjecture states that $J_G$ is Cohen-Macaulay if and only if $J_G$ is unmixed and the collection of the cut sets of $G$ is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to $12$ vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to $15$ vertices and all blocks with whiskers where the block has at most $11$ vertices. This significantly extends previous computational results.