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

  • SARIKA DEVHARE (a1), VINAYAK JOSHI (a2) and JOHN LAGRANGE (a3)
Abstract

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.

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vvjoshi@unipune.ac.in
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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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[1] 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.
[2] Anderson, D. D. and Naseer, M., ‘Beck’s coloring of a commutative ring’, J. Algebra 159 (1993), 500514.
[3] Anderson, D. F. and Livingston, P. S., ‘The zero-divisor graph of a commutative ring’, J. Algebra 217 (1999), 434447.
[4] Beck, I., ‘Coloring of a commutative ring’, J. Algebra 116 (1988), 208226.
[5] Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings I’, J. Algebra Appl. 10(4) (2011), 727739.
[6] Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings II’, J. Algebra Appl. 10(4) (2011), 741753.
[7] Bollobás, B., Modern Graph Theory (Springer, New York, 1998).
[8] 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), 119125.
[9] Chakrabarty, I., Ghosh, S., Mukherjee, T. and Sen, M., ‘Intersection graphs of ideals of rings’, Discrete Math. 309 (2009), 53815392.
[10] Csákány, B. and Pollák, G., ‘The graph of subgroups of a finite group’, Czechoslovak Math. J. 19 (1969), 241247.
[11] Davey, B. A. and Priestley, H. A., Introduction to Lattices and Order (Cambridge University Press, Cambridge, 2002).
[12] DeMeyer, F. R., McKenzie, T. and Schneider, K., ‘The zero-divisor graph of a commutative semigroup’, Semigroup Forum 65 (2002), 206214.
[13] Dilworth, R., ‘A decomposition theorem for partially ordered sets’, Ann. Math. (2) 51 (1950), 161166.
[14] Erdős, P., Goodman, A.W. and Pósa, L., ‘The representation of a graph by set intersections’, Canad. J. Math. 18(1) (1966), 106112.
[15] Halaš, R. and Jukl, M., ‘On Beck’s coloring of partially ordered sets’, Discrete Math. 309 (2009), 45844589.
[16] Joshi, V., ‘Zero divisor graph of a partially ordered set with respect to an ideal’, Order 29(3) (2012), 499506.
[17] Joshi, V. and Khiste, A., ‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177190.
[18] LaGrange, J. D. and Roy, K. A., ‘Poset graphs and the lattice of graph annihilators’, Discrete Math. 313(10) (2013), 10531062.
[19] Lam, T. Y., Lectures on Modules and Rings (Springer, New York, 1998.).
[20] Lovász, L., ‘Normal hypergraphs and the perfect graph conjecture’, Discrete Math. 2 (1972), 253267.
[21] Lu, D. and Wu, T., ‘The zero-divisor graphs of partially ordered sets and an application to semigroups’, Graph Combin. 26 (2010), 793804.
[22] Mirsky, L., ‘A dual of Dilworth’s decomposition theorem’, Amer. Math. Monthly 78 (1971), 876877.
[23] Patil, A., Waphare, B. N. and Joshi, V., ‘Perfect zero-divisor graphs’, Discrete Math. 340(4) (2017), 740745.
[24] Redmond, S. P., ‘The zero-divisor graph of a noncommutative ring’, Int. J. Commut. Rings 1(4) (2002), 203211.
[25] 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), 307320.
[26] Zelinka, B., ‘Intersection graphs of finite abelian groups’, Czechoslovak Math. J. 25 (1975), 171174.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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