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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).

References

Akbari, S., Tavallaee, A. and Khalashi Ghezelahmad, S., ‘On the complement of the intersection graph of submodules of a module’, J. Algebra Appl. 14(8) (2015), Article ID 1550116.Google Scholar
Anderson, D. D. and Naseer, M., ‘Beck’s coloring of a commutative ring’, J. Algebra 159 (1993), 500–514.Google Scholar
Anderson, D. F. and Livingston, P. S., ‘The zero-divisor graph of a commutative ring’, J. Algebra 217 (1999), 434–447.Google Scholar
Beck, I., ‘Coloring of a commutative ring’, J. Algebra 116 (1988), 208–226.Google Scholar
Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings I’, J. Algebra Appl. 10(4) (2011), 727–739.Google Scholar
Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings II’, J. Algebra Appl. 10(4) (2011), 741–753.Google Scholar
Bollobás, B., Modern Graph Theory (Springer, New York, 1998).Google Scholar
Bosak, J., ‘The graphs of semigroups’, in: Theory of Graphs and its Applications: Proceedings of the Symposium held in Smolenice in June 1963 (Academic Press, New York, 1964), 119–125.Google Scholar
Chakrabarty, I., Ghosh, S., Mukherjee, T. and Sen, M., ‘Intersection graphs of ideals of rings’, Discrete Math. 309 (2009), 5381–5392.Google Scholar
Csákány, B. and Pollák, G., ‘The graph of subgroups of a finite group’, Czechoslovak Math. J. 19 (1969), 241–247.Google Scholar
Davey, B. A. and Priestley, H. A., Introduction to Lattices and Order (Cambridge University Press, Cambridge, 2002).Google Scholar
DeMeyer, F. R., McKenzie, T. and Schneider, K., ‘The zero-divisor graph of a commutative semigroup’, Semigroup Forum 65 (2002), 206–214.Google Scholar
Dilworth, R., ‘A decomposition theorem for partially ordered sets’, Ann. Math. (2) 51 (1950), 161–166.Google Scholar
Erdős, P., Goodman, A.W. and Pósa, L., ‘The representation of a graph by set intersections’, Canad. J. Math. 18(1) (1966), 106–112.Google Scholar
Halaš, R. and Jukl, M., ‘On Beck’s coloring of partially ordered sets’, Discrete Math. 309 (2009), 4584–4589.Google Scholar
Joshi, V., ‘Zero divisor graph of a partially ordered set with respect to an ideal’, Order 29(3) (2012), 499–506.Google Scholar
Joshi, V. and Khiste, A., ‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177–190.Google Scholar
LaGrange, J. D. and Roy, K. A., ‘Poset graphs and the lattice of graph annihilators’, Discrete Math. 313(10) (2013), 1053–1062.Google Scholar
Lam, T. Y., Lectures on Modules and Rings (Springer, New York, 1998.).Google Scholar
Lovász, L., ‘Normal hypergraphs and the perfect graph conjecture’, Discrete Math. 2 (1972), 253–267.Google Scholar
Lu, D. and Wu, T., ‘The zero-divisor graphs of partially ordered sets and an application to semigroups’, Graph Combin. 26 (2010), 793–804.Google Scholar
Mirsky, L., ‘A dual of Dilworth’s decomposition theorem’, Amer. Math. Monthly 78 (1971), 876–877.Google Scholar
Patil, A., Waphare, B. N. and Joshi, V., ‘Perfect zero-divisor graphs’, Discrete Math. 340(4) (2017), 740–745.Google Scholar
Redmond, S. P., ‘The zero-divisor graph of a noncommutative ring’, Int. J. Commut. Rings 1(4) (2002), 203–211.Google Scholar
Visweswaran, S. and Patel, H. D., ‘On the clique number of the complement of the annihilating ideal graph of a commutative ring’, Beitr. Algebra Geom. 57 (2016), 307–320.Google Scholar
Zelinka, B., ‘Intersection graphs of finite abelian groups’, Czechoslovak Math. J. 25 (1975), 171–174.Google Scholar