A remark on Hamilton cycles with few colors

Abstract

Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of $K_n$ with $n$ colours contains a Hamilton cycle with $\le O(\log n)$ colours. They proved that there is always a Hamilton cycle with $\le 8\sqrt{n}$ colours. In this note we improve this bound to $O({\log }^{3}n)$.