Chromatic Automorphisms of Consecutive Digraphs

Abstract.

 Let 1≤d≤n be two positive integers, e,h,c∈ℤ _n , and assume that the set Δ={e,e+h,…,e+(d−1)h}⊆ℤ _n has cardinality d. A consecutive digraph Γ is defined by taking ℤ _n as the set of vertices and the pairs (x,cx+a), with a∈Δ, as arcs. If gcd(c,n)=1, the digraph Γ is d-regular, and the arc (x,cx+a) is said to be of color a. An automorphism f of Γ is said to be a permuting colors automorphism if there exists a permutation σ of the set Δ of colors such that if (x,y) is an arc of color a then (f(x),f(y)) is an arc of color σ(a). If σ is the identity, then the automorphism f is called a preserving colors automorphism. In this paper, explicit description of the groups of preserving and permuting colors automorphisms are given. The action of these groups on the vertex set is also studied and sufficient conditions for these digraphs to be Cayley digraphs are derived.