Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs

TL;DR

This paper explores the Cohen-Macaulay property of binomial edge ideals, linking graph girth and structure to algebraic properties, and simplifies characterization through initial ideals and graph decomposition.

Contribution

It introduces new criteria for Cohen-Macaulay binomial edge ideals based on graph structure and initial ideals, simplifying their characterization.

Findings

Cohen-Macaulay binomial edge ideals relate to graphs with girth less than 5 or infinite.

Characterization reduces to biconnected graphs with whiskers.

Provides necessary and sufficient conditions based on graph blocks.

Abstract

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only "biconnected graphs with some whisker attached" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or equal to infinity.