Almost Cohen-Macaulay bipartite graphs and 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) ≤ 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 \ {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.