On proper hamiltonicity and proper (even) pancyclicity of arc-colored complete (balanced bipartite) digraphs

Abstract

A subdigraph of an arc-colored digraph is called properly colored if its every pair of consecutive arcs have distinct colors. We call an arc-colored digraph D properly hamiltonian if it contains a properly colored Hamilton cycle, and properly (even) pancyclic if it contains a properly colored cycle of length k for every (even) k with $2\le k\le \left|V\right(D\left)\right|$. In this paper, we first obtain some color number conditions for the existence of properly colored Hamilton cycles of arc-colored complete (balanced bipartite) digraphs, and further prove that the these conditions can still guarantee the (even) pancyclicity of arc-colored complete (balanced bipartite) digraphs.