Abstract
We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in Q. J. Pure Appl. Math. 24:332–338, 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 even if the graph is a for forests of paths of arbitrary lengths (for any r≥2, provided \(s < 2r - \sqrt{2r + \frac{1}{4}}+ \frac{3}{2}\)). Furthermore we obtain a complete characterization of the problem’s complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any r≥2, if 3≤s≤2r−1 (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if s<7 for r=2, and s<6r−3, for r≥3. The result for planar graphs also 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.
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References
Beineke, L.W.: Biplanar graphs: a survey. Comput. Math. Appl. 34(11), 1–8 (1997)
Beineke, L.W., Harary, F.: The thickness of the complete graph. Can. J. Math. 17, 850–859 (1965)
Beineke, L.W., Harary, F., Moon, J.W.: On the thickness of the complete bipartite graph. Proc. Camb. Philos. Soc. 60(1), 1–5 (1964)
Brooks, R.L.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37, 194–197 (1941)
Bryant, D.E.: Cycle decompositions of the complete graphs. In: Hilton, A.J.W., Talbot, J.M. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Notes Series, vol. 346, pp. 67–97. Cambridge University Press, Cambridge (2007)
Cooper, C., McGrae, A.R.A., Zito, M.: Martingales on trees and the empire chromatic number of random trees. In: Kutyłowski, M., Gebala, M., Charatonik, W. (eds.) FCT 2009. Lecture Notes in Computer Science, vol. 5699, pp. 74–83. Springer, Berlin (2009)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (1999)
Garey, M.R., Johnson, D.S.: Computer and Intractability, a Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Gibbons, A.M.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)
Heawood, P.J.: Map colour theorem. Q. J. Pure Appl. Math. 24, 332–338 (1890)
Hutchinson, J.P.: Coloring ordinary maps, maps of empires, and maps of the moon. Math. Mag. 66(4), 211–226 (1993)
Jackson, B., Ringel, G.: Solution of Heawood’s empire problem in the plane. J. Reine Angew. Math. 347, 146–153 (1983)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations. Proceedings of a Symposium on the Complexity of Computer Computations, March 1972, Yorktown Heights, NY, pp. 85–103. Plenum Press, New York (1972)
Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. North-Holland, Amsterdam (1993)
Lucas, E.: Récreations Mathématiqués, vol. II. Gauthier-Villars, Paris (1892)
Mäkinen, E., Poranen, T.: An annotated bibliography on the thickness, outerthickness, and arboricity of a graph. In: D-NET PUBLICATIONS, vol. D-2009-3. University of Tampere, Tampere (2009)
McGrae, A.R., Zito, M.: Colouring random empire trees. In: Ochmański, E., Tyszkiewicz, J. (eds.) Mathematical Foundations of Computer Science 2008. Lecture Notes in Computer Science, vol. 5162, pp. 515–526. Springer, Berlin (2008)
McGrae, A.R.A.: Colouring empires in random trees. PhD thesis, Department of Computer Science, University of Liverpool (2010). Available from http://www.csc.liv.ac.uk/research/techreports/techreports.html as technical report ULCS-10-007
Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Comb. 14, 59–73 (1998)
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A preliminary version of this work appeared in the Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science 2011 (on pp. 179–190 of the book published by Springer as volume 6986 of the Lecture Notes in Computer Science Series).
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McGrae, A.R.A., Zito, M. The Complexity of the Empire Colouring Problem. Algorithmica 68, 483–503 (2014). https://doi.org/10.1007/s00453-012-9680-0
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DOI: https://doi.org/10.1007/s00453-012-9680-0