Mathematics > Combinatorics
[Submitted on 21 Oct 2022 (v1), last revised 24 Sep 2024 (this version, v3)]
Title:Short rainbow cycles for families of matchings and triangles
View PDF HTML (experimental)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.
Submission history
From: He Guo [view email][v1] Fri, 21 Oct 2022 20:48:12 UTC (8 KB)
[v2] Sun, 20 Nov 2022 20:35:04 UTC (8 KB)
[v3] Tue, 24 Sep 2024 10:50:34 UTC (11 KB)
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