When the Arc-Colored Line Digraph of a Cayley Colored Digraph is Again a Cayley Colored Digraph

Let \(D_\Delta(G)\) be the Cayley colored digraph of a finite group \(G\) generated by \(\Delta\). The arc-colored line digraph of a Cayley colored digraph is obtained by appropriately coloring the arcs of its line digraph. In this paper, it is shown that the group of automorphisms of \(D_\Delta(G)\) that act as permutations on the color classes is isomorphic to the semidirect product of \(G\) and a particular subgroup of \(Aut\;G\). Necessary and sufficient conditions for the arc-colored line digraph of a Cayley colored digraph also to be a Cayley colored digraph are then derived.