Proper connection and proper-walk connection of digraphs

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the contains a directed path (a walk) from u to v such that arcs adjacent on (on have distinct colors. The proper connection number pc→(D) (the proper-walk wc→(D)) minimum colours make (properly-walk connected). We prove pc→(Cn(S))≤2 every circulant Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Furthermore, we give some sufficient conditions Hamiltonian satisfy pc→(D)=wc→(D)=2.