On color-automorphism vertex transitivity of semigroups

In this paper, we continue the study of Kelarev and Praeger devoted to the color-automorphism vertex transitivity of Cayley graphs of semigroups and we generalize and complete some of their results. For this purpose, first we show that for a semigroup S and a non-empty subset C⊆S, the ColAutC(S)-vertex-transitivity of Cay(S,C) is equivalent to the ColAut〈C〉(S)-vertex transitivity of Cay(S,〈C〉), where 〈C〉 denotes the subsemigroup generated by C in S. Then we use this result to characterize a color-automorphism vertex transitive Cayley graph Cay(S,C), where for every a∈S, 〈C〉a is a simple 〈C〉-act or for every a∈S, 〈C〉a is finite. Similarly, we characterize a ColAutC(S)-vertex-transitive Cay(S,C) when for every c∈C, |〈c〉| is infinite and c is left cancellable. Finally, we use these results to establish that if S=∪̇α∈YSα is a semilattice of semigroups Sα and C is a non-empty subset of S, then the ColAutC(S)-vertex-transitivity of Cay(S,C) implies that Y has an identity e and C=Ce. This answers an open question asked in a previous article.