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Cohen-Macaulay bipartite graphs in arbitrary codimension


Authors: Hassan Haghighi, Siamak Yassemi and Rahim Zaare Nahandi
Journal: Proc. Amer. Math. Soc. 143 (2015), 1981-1989
MSC (2010): Primary 13H10, 05C75
DOI: https://doi.org/10.1090/S0002-9939-2015-12433-1
Published electronically: January 14, 2015
MathSciNet review: 3314108
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Abstract: Let $ G$ be an unmixed bipartite graph of dimension $ d-1$. Assume that $ K_{n,n}$, with $ n\ge 2$, is a maximal complete bipartite subgraph of $ G$ of minimum dimension. Then $ G$ is Cohen-Macaulay in codimension $ t$ if and only if $ 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$ which is Cohen-Macaulay in codimension $ t$, is obtained from a Cohen-Macaulay graph by replacing certain edges of $ G$ with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.


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Additional Information

Hassan Haghighi
Affiliation: Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Email: haghighi@kntu.ac.ir

Siamak Yassemi
Affiliation: School of Mathematics, Statistics & Computer Science, University of Tehran, Tehran Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: yassemi@ipm.ir, yassemi@ut.ac.ir

Rahim Zaare Nahandi
Affiliation: School of Mathematics, Statistics & Computer Science, University of Tehran, Tehran Iran
Email: rahimzn@ut.ac.ir

DOI: https://doi.org/10.1090/S0002-9939-2015-12433-1
Keywords: Flag complex, Cohen-Macaulay complex in codimension
Received by editor(s): March 19, 2013
Received by editor(s) in revised form: December 15, 2013
Published electronically: January 14, 2015
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society