Graph Colouring with No Large Monochromatic Components

Abstract

For a graphGand an integertwe letmcct(G) be the smallestmsuch that there exists a colouring of the vertices ofGbytcolours with no monochromatic connected subgraph having more thanmvertices. Letbe any non-trivial minor-closed family of graphs. We show thatmcc2(G) =O(n2/3) for anyn-vertex graphG∈. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such, and every fixedtwe show thatmcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphsGwith noKt+3minor and withmcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos provedmcc2(G) ≤ 20000 for every graphGof maximum degree 5. We show that there aren-vertex 7-regular graphsGwithmcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there existscϵ> 0 andn-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and withmcc2(G) ≥cϵn. For 6-regular graphs it is known only that the maximum order of magnitude ofmcc2is between$\sqrt n$andn.We also offer a Ramsey-theoretic perspective of the quantitymcct(G).