  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.
