Digraphs with proper connection number two

Abstract

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc→(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc→(D)=1 if and only if D is complete. Magnant et al. showed that pc→(D)≤3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).