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

TL;DR

The work addresses the Erdős–Gyárfás conjecture on monochromatic path covers in 2-edge-coloured complete graphs, aiming to reduce the required number of same-colour paths from $2\sqrt{n}$ to $\sqrt{n}$. It develops a bootstrap inductive framework that first establishes a bound $f(n)<\sqrt{n}+C$ for a large constant $C$, and then leverages a long monochromatic path together with bipartite-decomposition lemmas to show that the remaining vertices can be covered with at most $\sqrt{n}$ additional paths of the same colour. The main contribution is an asymptotic resolution of the conjecture for all sufficiently large $n$ (with explicit constants, e.g. $n>20^{40}$, where the bound holds), and the introduction of auxiliary bipartite Ramsey-type tools and path-decomposition techniques that may inform further refinements. This advances the understanding of monochromatic structures in edge-coloured complete graphs and connects with related partitioning results such as Lehel’s conjecture and its extensions.

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 $n$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof