Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
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 $K_n$ 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 $K_n$ 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 $K_n$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2016
- DOI:
- 10.48550/arXiv.1608.07028
- arXiv:
- arXiv:1608.07028
- Bibcode:
- 2016arXiv160807028A
- Keywords:
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- Mathematics - Combinatorics;
- 05C38;
- 05C45;
- 05B15
- E-Print:
- 9 pages, 1 figure