On s-t paths and trails in edge-colored graphs

In this paper we deal from an algorithmic perspective with different questions regarding monochromatic and properly edge-colored s-t paths/trails on edge-colored graphs. Given a c-edge-colored graph G c without properly edge-colored closed trails, we present a polynomial time procedure for the determination of properly edge-colored s-t trails visiting all vertices of G c a prescribed number of times. As an immediate consequence, we polynomially solve the Hamiltonian path (resp., Eulerian trail) problem for this particular class of graphs. In addition, we prove that to check whether G c contains 2 properly edge-colored s-t paths/trails with length at most L > 0 is NP-complete in the strong sense. Finally, we prove that, if G c is a general c-edge-colored graph, to find 2 monochromatic vertex disjoint s-t paths with different colors is NP-complete.