On Rainbow Cycles and Paths

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of Kn, there is a rainbow path on (3/4− o(1))n vertices, improving on the previously best bound of (2n + 1)/3 from [?]. Similarly, a k-rainbow path in a proper edge coloring of Kn is a path using no color more than k times. We prove that in every proper edge coloring of Kn, there is a k-rainbow path on (1− 2/(k + 1)!)n vertices.