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Colouring planar graphs with three colours and no large monochromatic components. (English) Zbl 1334.05030
The main result of the paper proves the existence of a function $$f :\mathbb{N} \to \mathbb{N}$$ such that the vertices of every planar graph with maximum degree $$\Delta$$ can be 3-coloured in such a way that each monochromatic component has at most $$f(\Delta)$$ vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by J. Kleinberg et al. in [“Storage management for evolving databases”, in: Proceedings of the 38th annual IEEE symposium on foundations of computer science (FOCS 1997), October 20–22, 1997, Miami Beach, Florida. Los Alamitos, CA: IEEE Computer Society. 353–362 (1997; doi:10.1109/SFCS.1997.646124)].
The result can be extended to a higher genus which improves a special case of the result of N. Alon et al. [J. Comb. Theory, Ser. B 87, No. 2, 231–243 (2003; Zbl 1023.05045)].

MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs 05C75 Structural characterization of families of graphs
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References:
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