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Sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs are inverstigated. In particular, we prove that an edgecolored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to n+1 2 has properly edge-colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge-colored paths and hamiltonian paths between given vertices are considered as well.

a b s t r a c tThis 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 edgecolored 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-edgecolored 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.