Short rainbow cycles for families of matchings and triangles

10/21/2022

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by   He Guo, et al.

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A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni <cit.>, is that any family ℱ=(F_1, …,F_n) of sets of edges in K_n, each of size k, has a rainbow cycle of length at most ⌈n/k⌉. In <cit.> 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.