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ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET

Part of: Ordered sets Graph theory

Published online by Cambridge University Press:  02 November 2017

SARIKA DEVHARE ,
VINAYAK JOSHI Open the ORCID record for VINAYAK JOSHI [Opens in a new window]  and
JOHN LAGRANGE
SARIKA DEVHARE
Affiliation:
Department of Mathematics, Savitribai Phule Pune University, Pune 411007, Maharashtra, India email sarikadevhare@gmail.com
VINAYAK JOSHI*
Affiliation:
Department of Mathematics, Savitribai Phule Pune University, Pune 411007, Maharashtra, India email vvjoshi@unipune.ac.in email vinayakjoshi111@yahoo.com
JOHN LAGRANGE
Affiliation:
Division of Natural Science and Mathematics, Lindsey Wilson College, Columbia, KY 42728, USA email lagrangej@lindsey.edu
*
✉ vvjoshi@unipune.ac.in
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Abstract

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In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

Keywords

zero-divisor graphintersection graphannihilating-ideal graphweakly perfect graph

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 06A07: Combinatorics of partially ordered sets 05C17: Perfect graphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 2 , April 2018 , pp. 185 - 193
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is financially supported by the University Grants Commission, New Delhi, via Senior Research Fellowship Award Letter No. F.17-37/2008(SA-I).

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