Counterexample to Conjectures on Complemented Zero-Divisor Graphs of Semigroups

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 Gᵣ(S) is isomorphic to the graph G(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.