Hamilton cycles in circulant digraphs with prescribed number of distinct jumps

Let G be a digraph that consists of a finite set of vertices V(G). G is called a circulant digraph if its automorphism group contains a | V ( G ) | -cycle. A circulant digraph G has arcs ( i , i + a 1 ) , ( i , i + a 2 ) , … , ( i , i + a j ) ( mod | V ( G ) | ) incident to each vertex i , where integers a k s satisfy 0 < a 1 < a 2 < a j ≤ | V ( G ) | − 1 and are called jumps. It is well known that not every G is Hamiltonian. In this paper we give sufficient conditions for a G to have a Hamilton cycle with prescribed distinct jumps, and prove that such conditions are also necessary for every G with two distinct jumps. Finally, we derive several results for obtaining G ′ with k , k ≥ 2 distinct jumps if the corresponding G contains a Hamilton cycle with two distinct jumps.