Short rainbow cycles for families of matchings and triangles

Abstract

A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family $F=\left(F_1,\dots ,F_n\right)$ of sets of edges in $K_n$, each of size $k$, has a rainbow cycle of length at most $\lceil \frac{n}{k}\rceil$. In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to $O\left(\log n\right)$ if all sets are matchings of size 2, or all are triangles. We show that the same is true in the mixed case, that is, if each $F_i$ is either a matching of size 2 or a triangle. We also study the case that each $F_i$ is a matching of size 2 or a single edge, or each $F_i$ is a triangle or a single edge, and in each of these cases we determine the threshold proportion between the types, beyond which the rainbow girth goes from linear to logarithmic.