Paths and trails in edge-colored graphs

Abstract

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 ( n 2 ) , 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.