Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$
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| Format: | Preprint |
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2024
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| _version_ | 1866929718142435328 |
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| author | Bandyopadhyay, Debmalya Lo, Allan |
| author_facet | Bandyopadhyay, Debmalya Lo, Allan |
| contents | Let $K_n^{(k)}$ be the complete $k$-graph on $n$ vertices. A $k$-uniform tight cycle is a $k$-graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge and any two consecutive edges share exactly $k-1$ vertices. A result of Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all $r$-edge coloured $K_{n}^{(k)}$ can be partitioned into $c_{r,k}$ vertex disjoint monochromatic tight cycles. However, the constant $c_{r,k}$ is of tower-type. In this work, we show that $c_{r, k}$ is a polynomial in $r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_17176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$ Bandyopadhyay, Debmalya Lo, Allan Combinatorics 05C65, 05C35 Let $K_n^{(k)}$ be the complete $k$-graph on $n$ vertices. A $k$-uniform tight cycle is a $k$-graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge and any two consecutive edges share exactly $k-1$ vertices. A result of Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all $r$-edge coloured $K_{n}^{(k)}$ can be partitioned into $c_{r,k}$ vertex disjoint monochromatic tight cycles. However, the constant $c_{r,k}$ is of tower-type. In this work, we show that $c_{r, k}$ is a polynomial in $r$. |
| title | Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$ |
| topic | Combinatorics 05C65, 05C35 |
| url | https://arxiv.org/abs/2408.17176 |