Random subgraphs of properly edgecoloured complete graphs and long rainbow cycles
Abstract
A subgraph of an edgecoloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edgecoloured 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 edgecoloured $K_n$ has a rainbow cycle of length $nO(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edgecoloured $K_n$ 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:

arXiv eprints
 Pub Date:
 August 2016
 DOI:
 10.48550/arXiv.1608.07028
 arXiv:
 arXiv:1608.07028
 Bibcode:
 2016arXiv160807028A
 Keywords:

 Mathematics  Combinatorics;
 05C38;
 05C45;
 05B15
 EPrint:
 9 pages, 1 figure