An Answer to a Question of Kelarev and Praeger on Cayley Graphs of Semigroups

In Europ. J. Combinatorics 24 (2003) 59--72, Kelarev and Praeger posed a question: Is it true that if $G$ is a semigroup with a subset $S$ such that Cay$(G,\{s\})$ is Aut16 February 2004In Europ. J. Combinatorics 24 (2003) 59--72, Kelarev and Praeger posed a question: Is it true that if $G$ is a semigroup with a subset $S$ such that Cay$(G,\{s\})$ is Aut$_{\{s\}}(G)$-vertex-transitive, for every $s\in S$, then the whole Cayley graph Cay$(G,S)$ is ColAut$_{S}(G)$-vertex-transitive, too? In this note, we give a negative answer to this problem and prove that in the cases of bands and completely simple semigroups the answer is positive.