Short rainbow cycles in sparse graphs

Abstract

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 \(\lceil \frac{n}{2} \rceil\), which is best possible. Our result settles a special case of a strengthening of the Caccetta-H\"aggkvist 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.