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Colouring Random Empire Trees

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Mathematical Foundations of Computer Science 2008 (MFCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5162))

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 first notice that 2r colours 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 ≥ 1, there exists a positive integer s r  < r such that, for large n, almost all n country r-empire trees need more than s r colours, and then to give lower bounds on the proportion of such maps that are colourable with s > s r colours.

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Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK

    Andrew R. McGrae & Michele Zito

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  1. Andrew R. McGrae
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  2. Michele Zito
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Edward Ochmański Jerzy Tyszkiewicz

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© 2008 Springer-Verlag Berlin Heidelberg

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McGrae, A.R., Zito, M. (2008). Colouring Random Empire Trees. In: Ochmański, E., Tyszkiewicz, J. (eds) Mathematical Foundations of Computer Science 2008. MFCS 2008. Lecture Notes in Computer Science, vol 5162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85238-4_42

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  • DOI: https://doi.org/10.1007/978-3-540-85238-4_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85237-7

  • Online ISBN: 978-3-540-85238-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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