Colour-biased Hamilton cycles in random graphs

TL;DR

This paper proves that in random graphs above the Hamiltonicity threshold, any edge r-colouring contains a Hamilton cycle with a nearly optimal proportion of edges of the same colour, extending understanding of colour-biased Hamilton cycles.

Contribution

It establishes the existence of nearly colour-balanced Hamilton cycles in random graphs with any edge colouring, achieving asymptotic optimality.

Findings

Existence of Hamilton cycles with at least (2/(r+1)-o(1))n same-colour edges.

Results hold for random graphs above the Hamiltonicity threshold.

The estimate is proven to be asymptotically optimal.

Abstract

We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This estimate is asymptotically optimal.