On $k$-uniform tight cycles: the Ramsey number for $C_{kn}^{(k)}$ and an approximate Lehel's conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915369161064448 |
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| author | Pfenninger, Vincent |
| author_facet | Pfenninger, Vincent |
| contents | A $k$-uniform tight cycle is a $k$-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. We show that, for each $k \geq 3$, the Ramsey number of the $k$-uniform tight cycle on $kn$ vertices is $(1+o(1))(k+1)n$. This is an extension to all uniformities of previous results for $k = 3$ by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for $k = 4$ by Lo and the author and confirms a special case of a conjecture by the former set of authors.
Lehel's conjecture, which was proved by Bessy and Thomassé, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for $k$-uniform tight cycles. We show that, for every $k \geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains a red tight cycle and a blue tight cycle that are vertex-disjoint and together cover $n - o(n)$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $k$-uniform tight cycles: the Ramsey number for $C_{kn}^{(k)}$ and an approximate Lehel's conjecture Pfenninger, Vincent Combinatorics 05D10, 05C65, 05C35 A $k$-uniform tight cycle is a $k$-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. We show that, for each $k \geq 3$, the Ramsey number of the $k$-uniform tight cycle on $kn$ vertices is $(1+o(1))(k+1)n$. This is an extension to all uniformities of previous results for $k = 3$ by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for $k = 4$ by Lo and the author and confirms a special case of a conjecture by the former set of authors. Lehel's conjecture, which was proved by Bessy and Thomassé, states that every red-blue edge-coloured complete graph contains a red cycle and a blue cycle that are vertex-disjoint and together cover all the vertices. We also prove an approximate version of this for $k$-uniform tight cycles. We show that, for every $k \geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains a red tight cycle and a blue tight cycle that are vertex-disjoint and together cover $n - o(n)$ vertices. |
| title | On $k$-uniform tight cycles: the Ramsey number for $C_{kn}^{(k)}$ and an approximate Lehel's conjecture |
| topic | Combinatorics 05D10, 05C65, 05C35 |
| url | https://arxiv.org/abs/2406.14468 |