A proof of a conjecture of Erd\H{o}s and Gy\'{a}rf\'{a}s on monochromatic path covers

TL;DR

This paper proves a long-standing conjecture by Erdős and Gyárfás, showing that for large enough graphs, the number of monochromatic paths needed to cover all vertices can be reduced from 2√n to √n.

Contribution

The authors confirm Erdős and Gyárfás's conjecture, demonstrating that the bound on monochromatic path covers can be improved for sufficiently large complete graphs.

Findings

Confirmed the conjecture for large n

Reduced the path cover number from 2√n to √n

Established a new bound for monochromatic path covers

Abstract

In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s 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 $n$.