A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914969438650368 |
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| author | Hollom, Lawrence |
| author_facet | Hollom, Lawrence |
| contents | An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a palette of $k$ colours. While it is not necessarily possible to find such a perfect matching, one can ask for a perfect matching as close to colour-balanced as possible. In particular, for a colouring $c:E(K_{2nk})\rightarrow [k]$, we seek to find a perfect matching $M$ minimising $f(M) = \sum_{i=1}^k\bigl||c^{-1}(i)\cap M|-n\bigr|$.
The previous best upper bound, due to Pardey and Rautenbach, was $\min f(M)\leq \mathcal{O}(k\sqrt{nk\log k})$. We remove the $n$-dependence, proving the existence of a matching $M$ with $f(M)\leq 4^{k^2}$ for all $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques Hollom, Lawrence Combinatorics 05C15 An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a palette of $k$ colours. While it is not necessarily possible to find such a perfect matching, one can ask for a perfect matching as close to colour-balanced as possible. In particular, for a colouring $c:E(K_{2nk})\rightarrow [k]$, we seek to find a perfect matching $M$ minimising $f(M) = \sum_{i=1}^k\bigl||c^{-1}(i)\cap M|-n\bigr|$. The previous best upper bound, due to Pardey and Rautenbach, was $\min f(M)\leq \mathcal{O}(k\sqrt{nk\log k})$. We remove the $n$-dependence, proving the existence of a matching $M$ with $f(M)\leq 4^{k^2}$ for all $k$. |
| title | A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2410.07993 |