Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

Abstract

Let $G$ be an Eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of $G$ is an Eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compatible circuits in edge-colored Eulerian undirected graphs. From our characterization for digraphs we develop a polynomial time algorithm that determines the existence of a compatible circuit in an edge-colored Eulerian digraph and produces a compatible circuit if one exists. Our results use the fact that rainbow spanning trees have been characterized in edge-colored undirected multigraphs. We provide another graph-theoretical proof of this fact.