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 \(\text {Cay}(G,C)\)]. 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.

1982年，Babai和Godsil推测几乎所有的Cayley有向图都是按字母顺序表示的。1998年，Xu推测几乎所有的Cayley有向图都是正常的[即\（G_L \）是\（\ text {Cay}（G，C）\）的自同构群的一个正常子群。]。最终，在1994年，Praeger和Mckay推测几乎所有无向的顶点传递图都是组的Cayley图。在本文中，首先，我们给出了针对id半群的Cayley有向图的这些猜想的变体，并确定了Babai和Godsil猜想的变体何时与Xu猜想的变体等效。然后，作为我们结果的一个特殊结果，我们得出结论，徐的猜想等同于巴拜和戈德西尔的猜想。另一方面，我们对Praeger和Mckay猜想的变体给出了肯定的答案，并且证明了当且仅当与群的Cayley有同构同构时，a半群的Cayley有向图才是顶点可传递的。最后，我们使用此特征来对Kelarev和Praeger提出的关于mono半群的Cayley有向图的顶点传递性的问题给出肯定的答案。同样使用此特征，我们明确确定了半同形的顶点传递Cayley有向图的自同构组。