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.

Compatible spanning circuits in edge-colored graphs

“…More recently, Carraher and Hartke [19] provided necessary and sufficient conditions for the existence of compatible Euler tours in arc-colored eulerian digraphs avoiding certain vertices of outdegree 3. In [20], the same authors further established necessary and sufficient conditions for the existence of compatible Euler tours in several specific families of arc-colored eulerian digraphs (the first half and the latter half of an arc can receive different colors), whose vertex set consists of vertices of outdegree and indegree 3.…”

“…In [9], the authors also claimed that a similar algorithm can be given for arc-colored eulerian digraphs, and they stated a necessary and sufficient condition for the existence of compatible Euler tours in arc-colored eulerian digraphs. However, Carraher and Hartke [19] noticed that the sufficient condition stated by Benkouar et al [9] is in fact not sufficient for the existence of compatible Euler tours in arc-colored eulerian digraphs, by giving the following small counterexample (see Figure 7.4). Moreover, so far, there is still no result on necessary and sufficient condition for the existence of compatible Euler tours in arc-colored eulerian digraphs (without any assumptions on the class of digraphs).…”

Compatible spanning circuits in edge-colored graphs

“…The problem of determining necessary and sufficient conditions for the existence of compatible Euler tours in eulerian digraphs with partition systems seems much harder. More recently, Carraher and Hartke [19] provided necessary and sufficient conditions for the existence of compatible Euler tours in arc-colored eulerian digraphs avoiding certain vertices of outdegree 3. In [20], the same authors further established necessary and sufficient conditions for the existence of compatible Euler tours in several specific families of arc-colored eulerian digraphs (the first half and the latter half of an arc can receive different colors), whose vertex set consists of vertices of outdegree and indegree 3.…”