Short rainbow cycles for families of matchings and triangles
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914955391926272 |
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| author | Guo, He |
| author_facet | Guo, He |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12243 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Short rainbow cycles for families of matchings and triangles Guo, He Combinatorics Discrete Mathematics 05C35, 05C38, 05D40 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. |
| title | Short rainbow cycles for families of matchings and triangles |
| topic | Combinatorics Discrete Mathematics 05C35, 05C38, 05D40 |
| url | https://arxiv.org/abs/2210.12243 |