Properly coloured Hamiltonian cycles in edge-coloured complete graphs

TL;DR

This paper proves that for large enough complete edge-coloured graphs with a maximum monochromatic degree less than about half the vertices, a properly coloured Hamiltonian cycle exists, confirming an asymptotic version of a longstanding conjecture.

Contribution

It establishes an asymptotic version of Bollobás and Erdős's conjecture, showing such Hamiltonian cycles exist under weaker conditions than previously known.

Findings

Properly coloured Hamiltonian cycles exist under the given conditions.

The result improves previous bounds by Alon and Gutin.

Confirms the conjecture asymptotically for large graphs.

Abstract

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $\Delta_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two adjacent edges have the same colour. In 1976, Bollob\'as and Erd\H{o}s conjectured that every $K_n^c$ with $\Delta_{\rm mon}(K_n^c) < \lfloor n/2 \rfloor$ contains a properly coloured Hamiltonian cycle. In this paper, we show that for any $\varepsilon > 0 $, there exists an integer $n_0$ such that every $K_n^c$ with $\Delta_{\rm mon}(K_n^c) < (1/2 - \varepsilon) n $ and $n \ge n_0$ contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin. Hence, the conjecture of Bollob\'as and Erd\H{o}s is true asymptotically.