Properly colored C₄→'s in arc-colored complete and complete bipartite digraphs

A subdigraph of an arc-colored digraph is called properly colored if its every consecutive arcs have distinct colors. Let D be a digraph. For a digraph H, let pc(D,H) be the minimum number such that every arc-colored digraph D^C with c(D)≥pc(D,H) contains a properly colored copy of H, where c(D) is the number of colors of D^C. Let K_n↔ and K_m,n↔ be the digraphs obtained from the complete graph K_n and the complete bipartite graph K_m,n respectively by replacing each edge uv with a pair of symmetric arcs (u,v) and (v,u); and let C_k→ be the directed cycle of length k. In this paper we determine pc(K_n↔,C₄→), pc(K_m,n↔,C₄→) and characterize the corresponding extremal arc-colorings of digraphs.