Cohen-Macaulay-ness in codimension for bipartite graphs

TL;DR

This paper investigates the Cohen-Macaulay property in bipartite graphs, providing a generalized characterization and describing how such graphs can be constructed from Cohen-Macaulay graphs by replacing edges with complete bipartite subgraphs.

Contribution

It generalizes previous characterizations of Cohen-Macaulay bipartite graphs and describes a construction method for graphs with this property.

Findings

Gives a formula for Cohen-Macaulay in codimension based on maximal complete bipartite subgraphs.

Shows that Cohen-Macaulay bipartite graphs can be obtained by replacing edges with complete bipartite graphs.

Provides examples illustrating the construction and properties.

Abstract

Let $G$ be an unmixed bipartite graph of dimension $d-1$. Assume that $K_{n,n}$, with $n\ge 2$, is a maximal complete bipartite subgraph of $G$ of minimum dimension. Then $G$ is Cohen-Macaulay in codimension $d-n+1$. This generalizes a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and a result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph $G$ which is Cohen-Macaulay in codimension $t$, is obtained from a Cohen-Macaulay graph by replacing certain edges of $G$ with complete bipartite graphs. We provide some examples.