Empires Make Cartography Hard: The Complexity of the Empire Colouring Problem

  • Andrew R. A. McGrae
  • Michele Zito
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)


We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly r > 1 countries each. We prove that the problem can be solved in polynomial time using s colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than s/r. However, if s ≥ 3, the problem is NP-hard for forests of paths of arbitrary lengths (if s < r) for trees (if r ≥ 2 and s < 2r) and arbitrary planar graphs (if s < 7 for r = 2, and s < 6r − 3, for r ≥ 3). The result for trees shows a perfect dichotomy (the problem is NP-hard if 3 ≤ s ≤ 2r − 1 and polynomial time solvable otherwise). The one for planar graphs proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than 6r − 3 colours graphs of thickness r ≥ 3.


Polynomial Time Planar Graph Complete Graph Average Degree Colour Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrew R. A. McGrae
    • 1
  • Michele Zito
    • 1
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

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