A solution to the 2/3 conjecture

TL;DR

This paper proves a long-standing vertex domination conjecture for 3-coloured complete graphs, showing a small vertex set can dominate at least two-thirds of the vertices in one colour, advancing combinatorial graph theory.

Contribution

It provides a proof of the 2/3 conjecture for vertex domination in 3-coloured complete graphs, utilizing and extending recent combinatorial techniques.

Findings

Confirmed the 2/3 conjecture for all n-vertex complete graphs with 3-edge-colourings.

Established that at most three vertices can dominate at least two-thirds of the graph in one colour.

Built upon and extended methods from recent related research.

Abstract

We prove a vertex domination conjecture of Erd\H os, Faudree, Gould, Gy\'arf\'as, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have at least 2n/3 neighbours in one of the colours. Our proof makes extensive use of the ideas presented in "A New Bound for the 2/3 Conjecture" by Kr\'al', Liu, Sereni, Whalen, and Yilma.