A Proof of the Conjecture on complemented zero-divisor graphs of semigroups

TL;DR

This paper proves a long-standing conjecture in the structure of complemented zero-divisor graphs of semigroups, showing that certain conditions imply the graph is isomorphic to a power set graph.

Contribution

It confirms the third conjecture by demonstrating the isomorphism under specified conditions, completing the open question.

Findings

Proved the third conjecture positively.

Established isomorphism to G(𝒫(n)) under given conditions.

Counterexamples provided for the first two conjectures.

Abstract

In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove the third conjecture positively, which has remained an open question until now. The third conjecture is stated as if $G(S)$ is uniquely complemented with the clique number $3$ or greater and has the property that every vertex has a unique complement, then the graph $G(S)$ is isomorphic to the graph $G(\mathcal{P}(n))$, where $n$ is the clique number of $G(S)$.