A Proof of the Conjecture on complemented zero-divisor graphs of semigroups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917415759118336 |
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| author | Khiste, Anagha Tarte, Ganesh Joshi, Vinayak |
| author_facet | Khiste, Anagha Tarte, Ganesh Joshi, Vinayak |
| contents | 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 $G(S)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof of the Conjecture on complemented zero-divisor graphs of semigroups Khiste, Anagha Tarte, Ganesh Joshi, Vinayak Combinatorics Primary 05C15, Secondary 06A12, 13A70 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 $G(S)$. |
| title | A Proof of the Conjecture on complemented zero-divisor graphs of semigroups |
| topic | Combinatorics Primary 05C15, Secondary 06A12, 13A70 |
| url | https://arxiv.org/abs/2604.15535 |