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COUNTEREXAMPLE TO CONJECTURES ON COMPLEMENTED ZERO-DIVISOR GRAPHS OF SEMIGROUPS

Part of: Graph theory Ordered sets

Published online by Cambridge University Press:  26 September 2025

ANAGHA KHISTE Open the ORCID record for ANAGHA KHISTE [Opens in a new window] ,
GANESH TARTE Open the ORCID record for GANESH TARTE [Opens in a new window]  and
VINAYAK JOSHI Open the ORCID record for VINAYAK JOSHI [Opens in a new window]
ANAGHA KHISTE
Affiliation:
Department of Applied Science and Humanities, https://ror.org/01f8qyw75 Indian Institute of Information Technology , Pune 410507, Maharashtra, India e-mail: avanikhiste@gmail.com
GANESH TARTE
Affiliation:
Department of Applied Sciences and Humanities, Pimpri Chinchwad College of Engineering, Pune 411044, Maharashtra, India e-mail: ganesh.tarte@pccoepune.org
VINAYAK JOSHI*
Affiliation:
Department of Mathematics, https://ror.org/044g6d731 Savitribai Phule Pune University , Pune 411007, Maharashtra, India e-mail: vvjoshi@unipune.ac.in
*
e-mail: vinayakjoshi111@yahoo.com
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Abstract

This paper is motivated by two conjectures proposed by Bender et al. [‘Complemented zero-divisor graphs associated with finite commutative semigroups’, Comm. Algebra 52(7) (2024), 2852–2867], which have remained open questions. The first conjecture states that if the complemented zero-divisor graph $ G(S) $ of a commutative semigroup $ S $ with a zero element has clique number three or greater, then the reduced graph $ G_r(S) $ is isomorphic to the graph $ G(\mathcal {P}(n)) $. The second conjecture asserts that if $ G(S) $ is a complemented zero-divisor graph with clique number three or greater, then $ G(S) $ is uniquely complemented. We construct a commutative semigroup $ S $ with a zero element that serves as a counter-example to both conjectures.

Keywords

clique numbercommutative semi-groupcomplemented graphreduced graphuniquely complemented graphzero-divisor graph

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 06A12: Semilattices

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 5
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

The third author is financially supported by the Science and Engineering Research Board (DST) via Project CRG/2022/002184.

References

Anderson, D. and Livingstone, P., ‘The zero-divisor graph of a commutative ring’, J. Algebra 217 (1999), 434–447.10.1006/jabr.1998.7840CrossRefGoogle Scholar
Beck, I., ‘Coloring of commutative rings’, J. Algebra 116 (1988), 208–226.CrossRefGoogle Scholar
Bender, C., Cappaert, P., DeCoste, R. and DeMeyer, L., ‘Complemented zero-divisor graphs associated with finite commutative semigroups’, Comm. Algebra 52(7) (2024), 2852–2867.CrossRefGoogle Scholar
DeMeyer, F. and DeMeyer, L., ‘Zero-divisor graphs of semigroups’, J. Algebra 283 (2005), 190–198.CrossRefGoogle Scholar
DeMeyer, F., McKenzie, T. and Schneider, K., ‘The zero-divisor graph of a commutative semigroup’, Semigroup Forum 65 (2002), 206–214.CrossRefGoogle Scholar
Joshi, V. and Khiste, A. U., ‘The zero-divisor graphs of Boolean posets’, Math. Slovaca 64(2) (2014), 511–519.CrossRefGoogle Scholar
LaGrange, J., ‘Complemented zero divisor graphs and Boolean rings’, J. Algebra 315 (2007), 600–611.10.1016/j.jalgebra.2006.12.030CrossRefGoogle Scholar