Cohen-Macaulay bipartite graphs in arbitrary codimension

Abstract

Let G G be an unmixed bipartite graph of dimension d − 1 d-1 . Assume that K n , n K_{n,n} , with n ≥ 2 n\ge 2 , is a maximal complete bipartite subgraph of G G of minimum dimension. Then G G is Cohen-Macaulay in codimension t t if and only if t ≥ d − n + 1 t\ge d-n+1 . This is derived from a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and generalizes a recent result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph G G which is Cohen-Macaulay in codimension t t , is obtained from a Cohen-Macaulay graph by replacing certain edges of G G with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.