Near rainbow Hamilton cycles in dense graphs

TL;DR

This paper proves that dense, properly edge-coloured graphs with a certain minimum degree contain near-rainbow Hamilton cycles, extending previous results from complete graphs to more general graphs.

Contribution

It establishes the existence of near-rainbow Hamilton cycles in dense, properly edge-coloured graphs with bounded colour multiplicity, generalizing earlier work on complete graphs.

Findings

Every such graph contains a Hamilton cycle with almost all distinct colours.

The bound of 1/8 on colour multiplicity is optimal.

The minimum degree condition is at least (1/2 + ε)n.

Abstract

Finding near-rainbow Hamilton cycles in properly edge-coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on $n$ vertices contains a Hamilton cycle with at least $n-\sqrt{2n}$ distinct colours. This result was improved to $n-O(\log^2 n)$ by Balogh and Molla in 2019. In this paper, we consider Anderson's problem for general graphs with a given minimum degree. We prove every globally $n/8$-bounded (i.e. every colour is assigned to at most $n/8$ edges) properly edge-coloured graph $G$ with $\delta(G) \geq (1/2+\varepsilon)n$ contains a Hamilton cycle with $n-o(n)$ distinct colours. Moreover, we show that the constant $1/8$ is best possible.