Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2411.18743 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912136153792512 |
|---|---|
| author | Peng, Danni Yan, Zhifei |
| author_facet | Peng, Danni Yan, Zhifei |
| contents | Finding near-rainbow Hamilton cycles in properly edge-coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on $n$ vertices contains a Hamilton cycle with at least $n-\sqrt{2n}$ distinct colours. This result was improved to $n-O(\log^2 n)$ by Balogh and Molla in 2019.
In this paper, we consider Anderson's problem for general graphs with a given minimum degree. We prove every globally $n/8$-bounded (i.e. every colour is assigned to at most $n/8$ edges) properly edge-coloured graph $G$ with $δ(G) \geq (1/2+\varepsilon)n$ contains a Hamilton cycle with $n-o(n)$ distinct colours. Moreover, we show that the constant $1/8$ is best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Near rainbow Hamilton cycles in dense graphs Peng, Danni Yan, Zhifei Combinatorics Finding near-rainbow Hamilton cycles in properly edge-coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on $n$ vertices contains a Hamilton cycle with at least $n-\sqrt{2n}$ distinct colours. This result was improved to $n-O(\log^2 n)$ by Balogh and Molla in 2019. In this paper, we consider Anderson's problem for general graphs with a given minimum degree. We prove every globally $n/8$-bounded (i.e. every colour is assigned to at most $n/8$ edges) properly edge-coloured graph $G$ with $δ(G) \geq (1/2+\varepsilon)n$ contains a Hamilton cycle with $n-o(n)$ distinct colours. Moreover, we show that the constant $1/8$ is best possible. |
| title | Near rainbow Hamilton cycles in dense graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.18743 |