Cohen-Macaulay bipartite graphs in arbitrary codimension

DOI10.1090/S0002-9939-2015-12433-1zbMATH Open1405.13045arXiv1302.0368OpenAlexW1971091220MaRDI QIDQ5179381

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Publication date: 19 March 2015

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Abstract: Let $G$ be an unmixed bipartite graph of dimension $d-1$. Assume that $K_{n,n}$, with $nge2$, 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.

Full work available at URL: https://arxiv.org/abs/1302.0368

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