Paths and trails in edge-colored graphs

This paper deals with the existence and search of Properly Edge-Colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s - t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest Properly Edge-Colored path/trail between s and t for some particular graphs and characterize edge-colored graphs without Properly Edge-Colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint Properly Edge-Colored s - t paths/trails in a c-edge-colored graph G(c) is NP-complete even for k = 2 and c = Omega(n(2)), where n denotes the number of vertices in G(c). Moreover, we prove that these problems remain NP-complete for c-colored graphs containing no Properly Edge-Colored cycles and c = S?(n). We obtain some approximation results for those maximization problems together with polynomial results for some particulars classes of edge-colored graphs.