Colouring Random Empire Trees

Authors:
Andrew R. Mcgrae;Michele Zito
Affiliations:
Department of Computer Science, University of Liverpool, Liverpool, UK L69 3BX;Department of Computer Science, University of Liverpool, Liverpool, UK L69 3BX
Venue:
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Year:
2008

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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 rcountries. We first notice that 2rcolours are necessary and sufficient to solve the problem in the worst-case. Then we define the notion of a random r-empire tree and, applying a method for enumerating spanning trees in a particular class of graphs, we find exact and asymptotic expressions for all central moments of the number of (balanced) s-colourings of such graphs. Such result in turns enables us to prove that, for each r茂戮驴 1, there exists a positive integer srrsuch that, for large n, almost all ncountry r-empire trees need more than srcolours, and then to give lower bounds on the proportion of such maps that are colourable with s srcolours.