When the arc-colored line digraph of a cayley colored digraph is again a cayley colored digraph

Abstract

Let D6(G) be the Cayley colored ugraph of a finite group G generated by A. The arc-colored line digraph of a Cayley colored di- graph ie obtained by appropriately coloring the arcs of its line digraph. In this paper it is shown that the group of automorphisms of D6 (G) that act as permutations on the color classes is isomorphic to the gemidi- rect product of G aird a particular subgroup of AutG. Neceeeary and sufficient conditions for the arc-colored une digraph of a Cayley colored digraph ?lso to be a Cayley colored digraph are then derived. 1. DpptNITIoNs AND TnnulNotocY For the sake of completeness we define in this section the main con- cepts used in the rest of the paper. A digraph D = (V, A) consists of a set V = V (D) of points called uertices and a set A = A(D) of pairs of different vertices called o,rcs. Thus, neither loops nor parallel arcs are allowed. The order of D is its number of vertices lyl and the size of D is its number of arcs lA1. If @, V) is an arc from o to y we say that e is adiacent to y and also that y is adjacent from r. A digraph is d-regular when each vertex is adjacent to and from exactly d others. A digraph is strongly connecfed when for any r,A eV there is a directed path from t to y. Let C = {rt, czi...,cd} be a set of elements called colors. An arc- coloring of a d-regular digraph D is a mapping from A(D) to C such that the d arcs from any vertex n are mapped to different colors, and likewise for the d arcs to c. An ?u iomorphism of a digraph ^D is a permutation of V that preserves adjacency. For an arc-colored digraph, the concept of color preservittg automorphism is more appropriate. A color preseruing automorphism $ of an arc-colored digraph must also preserve the arc colors, that is, if (u, r) is an arc colored cit the arc (d(), d(r)) must also be colored ci. Alternatively, if for a € V and c € Cwe represent by uOc the vertex adjacent from u by an arc colored c, we can say that ,/ must satisfy