Short rainbow cycles in sparse graphs

Matt DeVos, Sebastián González Hermosillo de la Maza, Krystal Guo, Daryl Funk, Tony Huynh, Bojan Mohar, Amanda Montejano

Published 2018-06-03Version 1

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.