Alon, N. and Pokrovskiy, Alexey and Sudakov, B. (2017) Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles. Israel Journal of Mathematics 222 (1), pp. 317-331. ISSN 0021-2172.
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Abstract
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured Kn has a rainbow cycle of length n−O(n^3/4). One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured Kn formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties.
Metadata
| Item Type: | Article |
|---|---|
| Additional Information: | The final publication is available at Springer via the link above. |
| School: | Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School |
| Depositing User: | Alexey Pokrovskiy |
| Date Deposited: | 12 Dec 2018 16:53 |
| Last Modified: | 02 Aug 2023 17:47 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/25438 |
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