Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

TL;DR

This paper proves that every properly edge-coloured complete graph contains a nearly Hamiltonian rainbow cycle of length close to n, using probabilistic methods to analyze subgraph expansion properties.

Contribution

It improves previous bounds by showing such graphs have rainbow cycles of length n minus a small order term, and introduces a novel probabilistic approach to analyze subgraph expansion.

Findings

Every properly edge-coloured K_n has a rainbow cycle of length n - O(n^{3/4})

Random subgraphs of properly edge-coloured K_n exhibit similar properties to truly random graphs

Such subgraphs have strong expansion properties

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.