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 ▫$(G,c)$▫ contains a rainbow cycle of ...length at most ▫$\lceil \frac{n}{2} \rceil$▫, 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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