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arXiv · 1909.09178

Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

Hannah Guggiari·Alex Scott
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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 $δ(G)\geq(1-\varepsilon_k)n$ 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 Gyárfas and Sárközy.

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Hannah Guggiari, Alex Scott. 2020-12-29. Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree. https://arxiv.org/abs/1909.09178

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