Almost Cohen-Macaulay bipartite graphs and connected in codimension two

TL;DR

This paper investigates the properties of almost Cohen-Macaulay bipartite graphs, establishing conditions on vertex degrees and connectivity, and characterizes when Ferrers graphs are almost Cohen-Macaulay based on their connectivity in codimension two.

Contribution

It provides new characterizations of almost Cohen-Macaulay bipartite graphs, including degree constraints and a criterion for Ferrers graphs based on connectivity in codimension two.

Findings

Vertices of positive degree in such graphs have degree at most two.

Removing a degree-one vertex preserves the almost Cohen-Macaulay property.

Ferrers graphs are almost Cohen-Macaulay if and only if they are connected in codimension two.

Abstract

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $\deg(v) \leq 2$. In particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus\{v\}$ is almost Cohen-Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen-Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.