Short rainbow cycles for families of matchings and triangles

TL;DR

This paper proves that in mixed families of matchings and triangles, the rainbow cycle length can be logarithmic in size, extending previous results from uniform cases and identifying thresholds for mixed configurations.

Contribution

It extends the logarithmic bound for rainbow cycle lengths to mixed families of matchings and triangles, and determines thresholds for different family compositions.

Findings

Rainbow cycles of logarithmic length exist in mixed families.

Threshold proportions determine when rainbow girth shifts from linear to logarithmic.

Results generalize previous uniform case findings.

Abstract

A generalization of the famous Caccetta--H\"aggkvist conjecture, suggested by Aharoni [Rainbow triangles and the Caccetta-H\"aggkvist conjecture, J. Graph Theory (2019)], is that any family $\mathcal{F}=(F_1, \ldots,F_n)$ 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 [Rainbow cycles for families of matchings, Israel J. Math. (2023)] and [Non-uniform degrees and rainbow versions of the Caccetta-H\"aggkvist conjecture, SIAM J. Discrete Math. (2023)] it was shown that asymptotically this can be improved to $O(\log n)$ if all sets are matchings of size 2, or all are triangles. We show that the same is true in the mixed case, i.e., 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.