Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
Noga Alon, Alexey Pokrovskiy, Benny Sudakov

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 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 . In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured has a rainbow cycle of length . One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured 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.
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