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Seven Region Maps on A Torus

Published online by Cambridge University Press:  03 November 2016

John Leech
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Extract

It is well known that any map on a torus can be coloured with seven colours so that no two contiguous regions are of the same colour, and that fewer than seven colours will not suffice as there exist maps of seven regions of which each one abuts on to each other. In this paper all such maps of seven regions are analysed and a simple method of constructing them is given.

Type
Research Article
Information
The Mathematical Gazette , Volume 39 , Issue 328 , May 1955 , pp. 102 - 105
Copyright
Copyright © The Mathematical Association 1955

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References

* e.g. D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie (Berlin, 1932), p. 296, (English translation, Geometry and the Imagination (New York, 1952), p. 335) ; W.W Rouse Ball, Mathematical Recreations and Essays (Revised Coxeter, London, 1939), p. 235 ; H. Martyn Cundy and Rollett, Mathematical Models (Oxford, 1952), p. 149; Ungar, “ On Diagrams Representing Maps ”, J London Math. Soc., 28 (1953), p. 342, first diagram. His second diagram is equivalent to the first diagram in Figure 3 of this paper.

* As suggested by J. M. Andreas (quoted by Coxeter in Rouse Ball, loc. cit.).