… once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge …
Kn, there is a rainbow path on (3/4 − o(1))n vertices, improving on the previously best bound …

… , we will focus on long rainbow paths and rainbow cycles in edge … a path easily by deleting
an arbitrary edge from a cycle, there is a rainbow k-cycle means that there is a rainbow k-path. …

… Consider a vertex vi on a rainbow cycle C = v1v2 ···vlv1. Each edge vix where x is not on C
must be colored with a color used on an edge of C (that is not incident to vi) as otherwise we …

… Consider a vertex vi on a rainbow cycle C = v1v2 ···vℓv1. Each edge vix where x is not on C
must be colored with a color used on an edge of C (that is not incident to vi) as otherwise we …

… on n vertices has a rainbow cycle on at least n − O ( n 3 ∕ 4 ) vertices, by showing that G has
a rainbow cycle on … colored complete graph has a Hamiltonian cycle in which at least n − O …

… rather than just path forests, on the other hand it only assumes K n ↔ to be properly … n
contains a rainbow cycle of length n − O ( n 3 ∕ 4 ) and our proof of Theorem 1.1 is based on their …

… a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the
existence of r-rainbow cycles for three different flip graphs on … of plane trees on an arbitrary …

… Finally, we find a rainbow Hamiltonian path P on V (G)\V (C) and absorb P … a rainbow cycle
of length n−1 by following a classical proof of Dirac theorem. Then we obtain a rainbow cycle …

… Our result on Colorful Path motivates a related question on Rainbow st-Connectivity: can
one show that Rainbow st-Connectivity does not admit an algorithm running in time ( 2 − ε ) k n …

… The proof will be by induction on n. Clearly the conclusion holds when n = 1. Suppose that n
… so that there are no rainbow cycles in X. We arrange for the sets of colors on the edges of G[…