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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, 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, 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

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