Almost Cohen–Macaulay bipartite graphs and connected in codimension two

Abstract

In this paper, we study almost Cohen–Macaulay bipartite graphs. In particular, we prove that if [Formula: see text] is an almost Cohen–Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of [Formula: see text]. In particular, if [Formula: see text] is an almost Cohen–Macaulay bipartite graph and [Formula: see text] is a vertex of degree one of [Formula: see text] and [Formula: see text] its adjacent vertex, then [Formula: see text] 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.