Cohen-Macaulay bipartite graphs in arbitrary codimension
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- by Hassan Haghighi, Siamak Yassemi and Rahim Zaare Nahandi PDF
- Proc. Amer. Math. Soc. 143 (2015), 1981-1989 Request permission
Corrigendum: Proc. Amer. Math. Soc. 149 (2021), 3597-3599.
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.References
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Additional Information
- Hassan Haghighi
- Affiliation: Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
- MR Author ID: 109235
- ORCID: 0000-0002-6962-3738
- 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
- MR Author ID: 352988
- Email: yassemi@ipm.ir, yassemi@ut.ac.ir
- Rahim Zaare Nahandi
- Affiliation: School of Mathematics, Statistics & Computer Science, University of Tehran, Tehran Iran
- MR Author ID: 211459
- ORCID: 0000-0002-9257-6554
- Email: rahimzn@ut.ac.ir
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3314108