Counter-example to Conjectures on Complemented Zero-Divisor Graphs of Semigroups

TL;DR

This paper constructs a specific commutative semigroup that serves as a counter-example to two open conjectures about the structure of complemented zero-divisor graphs, challenging previous assumptions in algebraic graph theory.

Contribution

The paper provides the first known counter-example to two conjectures regarding complemented zero-divisor graphs of semigroups, disproving their general validity.

Findings

Counter-example disproves both conjectures

Shows that the conjectures do not hold universally

Challenges existing beliefs in algebraic graph theory

Abstract

In this paper, we are motivated by two conjectures proposed by C. Bender et al.\ in 2024, 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 the 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 the clique number three or greater, then \( G(S) \) is uniquely complemented. In this work, we construct a commutative semigroup \( S \) with a zero element that serves as a counter-example to both conjectures.