$(S_2)$-condition and Cohen-Macaulay binomial edge ideals

TL;DR

This paper investigates the properties of binomial edge ideals, linking the algebraic $(S_2)$ condition to graph accessibility, and characterizes Cohen-Macaulay cases among small graphs using combinatorial and computational methods.

Contribution

It introduces a simplicial complex framework to relate the $(S_2)$ condition to graph accessibility and characterizes Cohen-Macaulay binomial edge ideals for graphs up to 12 vertices.

Findings

If $J_G$ is $(S_2)$, then $G$ is accessible.

Characterization of accessible blocks with whiskers for cycle rank 3.

Graphs with ≤12 vertices and Cohen-Macaulay $J_G$ are exactly the accessible ones.

Abstract

We describe the simplicial complex $\Delta$ such that the initial ideal of $J_G$ is the Stanley-Reisner ideal of $\Delta$. By $\Delta$ we show that if $J_G$ is $(S_2)$ then $G$ is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen-Macaulay are all and only the accessible ones.