A remark on Hamilton cycles with few colors

TL;DR

This paper improves the upper bound on the number of colors needed in a Hamilton cycle within properly edge-colored complete graphs from 8√n to O(log^3 n), advancing understanding of color-efficient Hamilton cycles.

Contribution

The paper refines the upper bound on the number of colors in Hamilton cycles from 8√n to O(log^3 n) in properly edge-colored complete graphs.

Findings

Established an improved upper bound of O(log^3 n) colors for Hamilton cycles.

Confirmed the conjecture that Hamilton cycles can be found with logarithmically many colors.

Enhanced previous bounds from 8√n to a polylogarithmic function.

Abstract

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