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The chromatic number of the sphere

  • Gustavus J. Simmons (a1)

Abstract

Erdös, Harary and Tutte have defined the chromatic number of the plane to be the least number of sets partitioning the plane such that no set contains two points at unit distance apart. By analogy, the chromatic number χ(S1), of the sphere, Sr of radius r is defined to be the least number of sets partitioning the surface of Sr such that no set contains two points at unit chordal distance apart. In this paper it is proven that and that this bound is best possible since .

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References

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Erdös, P., Harary, F. and Tutte, W. T. (1965), ‘On the dimension of a graph’, Mathematika 12, 118122.
Hadwiger, H., Debrunner, H. and Klee, V. (1964), Combinatorial Geometry in the Plane, Holt, Rinehart & Winston, New York.
Moser, L. and Moser, W. (1961), ‘Problem and solution P10’, Canad. Math. Bull. 4, 187189.
Simmons, G. J. (1974), On a problem of Erdös concerning a 3-coloring of the unit sphere, Discrete Math. 8, 8184.
Woodall, D. R., (1973), ‘Distances realized by sets covering the plane’, J. Combinatorial Theory Series A 14, 187200.
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The chromatic number of the sphere

  • Gustavus J. Simmons (a1)

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