Abstract
We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps whose dual planar graph is a tree, with empires formed by exactly r countries. We prove that, for each fixed r > 1, with probability approaching one as the size of the graph grows to infinity, the minimum number of colours for which a feasible solution exists takes one of seven possible values.
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Cooper, C., McGrae, A.R.A., Zito, M. (2009). Martingales on Trees and the Empire Chromatic Number of Random Trees. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds) Fundamentals of Computation Theory. FCT 2009. Lecture Notes in Computer Science, vol 5699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03409-1_8
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DOI: https://doi.org/10.1007/978-3-642-03409-1_8
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