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Regularity and projective dimension of the edge ideal of $ C_5$-free vertex decomposable graphs


Authors: Fahimeh Khosh-Ahang and Somayeh Moradi
Journal: Proc. Amer. Math. Soc. 142 (2014), 1567-1576
MSC (2010): Primary 13D02, 13P10; Secondary 13D40, 13A02
Published electronically: February 17, 2014
MathSciNet review: 3168464
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Abstract: In this paper, we explain the regularity, projective dimension and depth of the edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $ C_5$-free vertex decomposable graph $ G$, $ \mbox {reg}(R/I(G))= c_G$, where $ c_G$ is the maximum number of $ 3$-disjoint edges in $ G$. Moreover, for this class of graphs we characterize $ \mbox {pd}(R/I(G))$ and $ \mbox {depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.


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Additional Information

Fahimeh Khosh-Ahang
Affiliation: Department of Mathematics, Ilam University, P. O. Box 69315-516, Ilam, Iran
Email: fahime{\textunderscore}khosh@yahoo.com

Somayeh Moradi
Affiliation: Department of Mathematics, Ilam University, P. O. Box 69315-516, Ilam, Iran; and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
Email: somayeh.moradi1@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11906-X
Keywords: Depth, edge ideal, projective dimension, regularity, vertex decom- posable
Received by editor(s): December 16, 2011
Received by editor(s) in revised form: June 20, 2012
Published electronically: February 17, 2014
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.