Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

TL;DR

This paper investigates monochromatic components in edge-coloured graphs with high minimum degree, establishing the maximum possible epsilon for 3-colourings and disproving a previous conjecture.

Contribution

It determines the maximum epsilon for which graphs with high minimum degree guarantee large monochromatic components in 3-colourings, disproving a prior conjecture.

Findings

Maximum epsilon for 3-colourings is 1/6.

Disproves Gyárfas and Sárközy's conjecture.

Provides bounds on monochromatic component sizes in high minimum degree graphs.

Abstract

For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gy\'{a}rfas and S\'{a}rk\"{o}zy.