Abstract
The Cayley graphs of some semigroups have been considered by many authors. Let S=⋃ α∈Y S α be a semilattice of semigroups and C⊆S. In this note, we investigate the color automorphism vertex-transitivity of the Cayley graph Cay(S,C), and as one of our main results, we show that under a natural assumption on this graph, the behavior of the Cayley graph Cay(S,C) is essentially related to the properties of the Cayley graph Cay(Y,D), where D={a∈Y|∃c∈C:c∈S a }. Because of this relation, first we study the Cayley graph Cay(T,D), where T is a semilattice and D⊆T, and we give some useful descriptions for semilattice graphs. Then we use these results to characterize the color automorphism vertex-transitive Cayley graphs of semilattices of semigroups under our assumption.
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The authors are highly grateful to the referee for very valuable suggestions and corrections, which have helped us to improve the text of this article.
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Communicated by Norman R. Reilly.
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Khosravi, B., Khosravi, B. On Cayley graphs of semilattices of semigroups. Semigroup Forum 86, 114–132 (2013). https://doi.org/10.1007/s00233-012-9384-4
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DOI: https://doi.org/10.1007/s00233-012-9384-4