On the automorphism groups of vertex-transitive Cayley digraphs of monoids

In 1982, Babai and Godsil conjectured that almost all Cayley digraphs are digraphical regular representations. In 1998, Xu conjectured that almost all Cayley digraphs are normal [i.e., G(L) is a normal subgroup of the automorphism group of Cay(G,C). Finally, in 1994, Praeger and Mckay conjectured that almost all undirected vertex-transitive graphs are Cayley graphs of groups. In this paper, first we present the variants of these conjectures for Cayley digraphs of monoids and we determine when the variant of Babai and Godsil's conjecture is equivalent to the variant of Xu's conjecture. Then, as a special consequence of our results, we conclude that Xu's conjecture is equivalent to Babai and Godsil's conjecture. On the other hand, we give affirmative answer to the variant of Praeger and Mckay's conjecture and we prove that a Cayley digraph of a monoid is vertex-transitive if and only if it is isomorphic to a Cayley digraph of a group. Finally, we use this characterization to give an affirmative answer to a question raised by Kelarev and Praeger about vertex-transitivity of Cayley digraphs of monoids. Also using this characterization, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids.