Long properly coloured cycles in edge-coloured graphs

TL;DR

This paper proves that large edge-coloured graphs with high minimum colour degree contain long properly coloured cycles, extending understanding of cycle existence under colour constraints.

Contribution

It establishes a new lower bound on the length of properly coloured cycles in graphs with high minimum colour degree.

Findings

Graphs with minimum colour degree above (1/2 + ε)n contain long properly coloured cycles.

The cycle length is at least the minimum of n and two-thirds of the minimum colour degree.

Results hold for sufficiently large graphs with arbitrary ε > 0.

Abstract

Let $G$ be an edge-coloured graph. The minimum colour degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any $\varepsilon >0$ and $n$ large, every edge-coloured graph $G$ with $\delta^c(G) \ge (1/2+\varepsilon)n$ contains a properly coloured cycle of length at least $\min\{ n , \lfloor 2 \delta^c(G)/3 \rfloor\}$.