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Coloring island maps

Published online by Cambridge University Press:  17 April 2009

Brad Jackson  and
Gerhard Ringel
Brad Jackson
Affiliation:
Department of Mathematics, Division of Natural Sciences, University of California, Santa Cruz, California 95064, USA.
Gerhard Ringel
Affiliation:
Department of Mathematics, Division of Natural Sciences, University of California, Santa Cruz, California 95064, USA.
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An empire map is a map where the set of countries is divided into disjoint subsets which we call empires. If one empire consists of a single country then the remaining map is called an empire island map, the single country being referred to as the ocean. If every empire consists of at most M countries then it is called an M-pire map.

It is proved that an M-pire map is colorable by 6M – 2 colors. Further there exists such a map which has 6M – 2 mutually adjacent empires.

If an M-pire island map has every country adjacent to the ocean, then it is colorable in 4M colors. Such a map exists with 4M mutually adjacent empires where every country is adjacent to the ocean.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 2 , April 1984 , pp. 151 - 165
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Gardner, Martin, “Mathematical games”, Sci. Amer. (1980), 14.CrossRefGoogle Scholar
[2]Heawood, P. J., “Map-colour theorem”, Quart. J. Pure Appl. Math. 24 (1890), 332–338.Google Scholar