Large Monochromatic Components of Small Diameter

TL;DR

This paper improves bounds on the size and diameter of monochromatic components in 3-edge-colored complete graphs, showing either a large diameter-three component or spanning diameter-four components.

Contribution

It refines Gyárfás's conjecture for the case of three colors, establishing tighter bounds on monochromatic component sizes and diameters.

Findings

Existence of a large monochromatic component with diameter at most three or spanning diameter-four components in 3-colorings.

Improved bounds from previous diameter four results for the case of three colors.

Provides structural insights into monochromatic components in edge-colored complete graphs.

Abstract

Gy\'arf\'as conjectured in 2011 that every $r$-edge-colored $K_n$ contains a monochromatic component of bounded ("perhaps three") diameter on at least $n/(r-1)$ vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of $r=3$: We show that in every $3$-edge-coloring of $K_n$ either there is a monochromatic component of diameter at most three on at least $n/2$ vertices or every color class is spanning and has diameter at most four.