A note on colour-bias Hamilton cycles in dense graphs

TL;DR

This paper extends recent results on colour-biased Hamilton cycles in dense graphs from 2-colourings to r-colourings, establishing the minimum degree threshold for their existence.

Contribution

It generalizes the known threshold for 2-colourings to the case of r-colourings, providing a broader understanding of colour-biased Hamilton cycles.

Findings

Determined the minimum degree threshold for r-colourings to contain colour-biased Hamilton cycles.

Extended previous 2-colour results to r-colour scenarios.

Contributed to the theory of coloured Hamilton cycles in dense graphs.

Abstract

Balogh, Csaba, Jing and Pluh\'ar recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.