Paths and trails in edge-colored graphs

This paper deals with the existence and search for 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 a particular class of 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 = Ω (n²), where n denotes the number of vertices in G^c. Moreover, we prove that these problems remain NP-complete for c-edge-colored graphs containing no properly edge-colored cycles and c = Ω (n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.