Partitioning a graph into monochromatic connected subgraphs

TL;DR

This paper proves two conjectures regarding partitioning graphs with high minimum degree into monochromatic connected subgraphs for two and three colours, extending known results from complete graphs.

Contribution

It establishes the validity of two conjectures by Bal and DeBiasio for graphs with large minimum degree in the cases of two and three colours.

Findings

Confirmed conjecture for two colours

Confirmed conjecture for three colours

Extended results from complete to high minimum degree graphs

Abstract

A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by Erd\H{o}s, Pyber and Gy\'arf\'as that states that there exists a partition into $r-1$ monochromatic connected subgraphs. We consider a variant of this problem, where the complete graph is replaced by a graph with large minimum degree, and prove two conjectures of Bal and DeBiasio, for two and three colours.