Hamiltonian problems in edge-colored complete graphs and eulerian cycles in edge-colored graphs : some complexity results

Abstract

In an edge-colored, we say that a path (cycle) is alternating if it has length at least 2 (3) and if any 2 adjacent edges of this path (cycle) have different colors. We give efficient algorithms for finding alternating factors with a minimum number of cycles and then, by using this result, we obtain polynomial algorithms for finding alternating Hamiltonian cycles and paths in 2-edge-colored complete graphs. We then show that some extensions of these results to k-edge-colored complete graphs, k ≥ 3, are NP-complete. related problems are proposed. Finally, we give a polynomial characterization of the existence of alternating Eulerian cycles in edge-colored graphs. Our proof is algorithmic and uses a procedure that finds a perfect matching in a complete k-partite graph.