Bipartite $S_2$ graphs are Cohen-Macaulay

TL;DR

This paper establishes that bipartite graphs whose independence complex's Stanley-Reisner ring satisfies Serre's condition S_2 are Cohen-Macaulay, extending known characterizations and exploring related graph classes.

Contribution

It proves the equivalence of S_2 and Cohen-Macaulay properties for bipartite graphs and extends this to chordal and cyclic graphs.

Findings

Bipartite S_2 graphs are Cohen-Macaulay

Extension of Herzog-Hibi characterization

Classification of cyclic graphs by S_2 condition

Abstract

In this paper we show that if the Stanley-Reisner ring of the simplicial complex of independent sets of a bipartite graph $G$ satisfies Serre's condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi carries over this family of bipartite graphs. We check that the equivalence of Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs and we classify cyclic graphs with respect to the condition $S_2$.