On monochromatic path covers conjecture of Erd\H{o}s--Gy\'arf\'as

Hangdi Chen; Yaojun Chen

On monochromatic path covers conjecture of Erd\H{o}s--Gy\'arf\'as

Combinatorics 2026-07-24 v1

Authors: Hangdi Chen , Yaojun Chen

Abstract

Erd\H{o}s and Gy\'arf\'as conjectured in 1995 that, in every red--blue edge-coloring of a complete graph $K_n$ , the vertex set can be covered by at most $\sqrt n$ monochromatic paths, all of the same color. Pokrovskiy, Versteegen and Williams (JCT-B, 2026) proved the conjecture for all sufficiently large $n$ . In this paper, by using minimal counterexample method, we confirm the conjecture completely.

Keywords

monochromatic path cover Erdos-Gyarfas conjecture complete graph edge-coloring minimal counterexample

Cite

@article{arxiv.2607.21915,
  title  = {On monochromatic path covers conjecture of Erd\H{o}s--Gy\'arf\'as},
  author = {Hangdi Chen and Yaojun Chen},
  journal= {arXiv preprint arXiv:2607.21915},
  year   = {2026}
}