Cohen-Macaulay-ness in codimension for bipartite graphs

2013/02/02 by Hassan Haghighi, Siamak Yassemi, Haghighi, Hassan +4
Mathematics · #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #Graph theory and applications #math.AC #math.CO

paper · pdf · doi:10.48550/arxiv.1302.0368

9 pages

arxiv created 2013/02/02 · arxiv updated 2013/02/05

Abstract

Let G be an unmixed bipartite graph of dimension d-1. Assume that K_n,n, with n≥ 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.

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