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arXiv · 2409.03623

A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers

Alexey Pokrovskiy·Leo Versteegen·Ella Williams
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Abstract

In 1995, Erdős and Gyárfás proved that in every $2$-edge-coloured complete graph on $n$ vertices, there exists a collection of $2\sqrt{n}$ monochromatic paths, all of the same colour, which cover the entire vertex set. They conjectured that it is possible to replace $2\sqrt{n}$ by $\sqrt{n}$. We prove this to be true for all sufficiently large $n$.

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Alexey Pokrovskiy, Leo Versteegen, Ella Williams. 2025-10-07. A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers. https://arxiv.org/abs/2409.03623

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