Properly colored subgraphs and rainbow subgraphs in edge-colorings with local constraints
Random Structures & Algorithms
Combinatorics, Probability and Computing
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An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H, the rainbow Turan number ex^*(n,H) is defined as the maximum number of edges in a properly edge-colored graph on n vertices with no rainbow copy of H. We study the rainbow Turan number of even cycles, and prove that for every fixed @e0, there is a constant C(@e) such that every properly edge-colored graph on n vertices with at least C(@e)n^1^+^@e edges contains a rainbow cycle of even length at most 2@?ln4-ln@eln(1+@e)@?. This partially answers a question of Keevash, Mubayi, Sudakov, and Verstraete, who asked how dense a graph can be without having a rainbow cycle of any length.