Short rainbow cycles for families of matchings and triangles
He Guo
TL;DR
Addressing the rainbow Caccetta–Häggkvist framework, the paper proves that when each color class is either a $2$-edge matching or a triangle, the rainbow girth is $O( ext{log }n)$. It analyzes three mixed regimes with single edges and identifies threshold proportions (e.g., $α>1/2$ for matchings and $α>0$ for triangles) above which the logarithmic bound holds, using a random-subset technique and Bollobás–Szemerédi-type girth bounds together with standard concentration inequalities. Sharpness results show the thresholds are tight: $α>1/2$ is necessary for the matchings case and $α>0$ for triangles. The work extends prior results for pure classes to mixed colorings, clarifying when short rainbow cycles exist and contributing to the broader understanding of rainbow analogues of CHC in extremal graph theory.
Abstract
A generalization of the famous Caccetta--Häggkvist conjecture, suggested by Aharoni [Rainbow triangles and the Caccetta-Häggkvist 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-Häggkvist 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.