An edge-colored directed graph is called properly connected if, between every pair of vertices, there is a properly colored directed path. We study some conditions on directed graphs which guarantee the existence of a coloring that is properly connected. We also study conditions on a colored directed graph which guarantee that the coloring is properly connected.
Magnant, Colton, Patrick R. Morley, Sarabeth A. Porter, Pouria Salehi Nowbandegani, Hua Wang.
"Directed Proper Connection of Graphs."
Matematički Vesnik, 68 (1): 58-65.