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On the chromatic number of plane tilings

Published online by Cambridge University Press:  09 April 2009

D. Coulson
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia e-mail: [email protected]
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Abstract

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It is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Coulson, D., ‘A 15-colouring of 3-space omitting distance one’, Discrete Math. 256 (2002), 8390.Google Scholar
[2]Raiskii, D. E., ‘The realisation of all distances in a decomposition of the space Rn into n + 1 parts’, Math. Notes 7 (1970), 194196.CrossRefGoogle Scholar