Short rainbow cycles in sparse graphs
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Let G be a simple n-vertex graph and c be a colouring of E(G) with n colours, where each colour class has size at least 2. We prove that G contains a rainbow cycle of length at most n/2, which is best possible. Our result settles a special case of a strengthening of the Caccetta-Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.
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