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

Abstract

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 $c\left(D\right)\ge pc(D,H)$ contains a properly colored copy of H, where $c\left(D\right)$ is the number of colors of $D^C$. Let $\overleftrightarrow{K_n}$ and $\overleftrightarrow{K_{m,n}}$ 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 $\overrightarrow{C_k}$ be the directed cycle of length k. In this paper we determine $pc(\overleftrightarrow{K_n},\overrightarrow{C_4})$, $pc(\overleftrightarrow{K_{m,n}},\overrightarrow{C_4})$ and characterize the corresponding extremal arc-colorings of digraphs.