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On the chromatic number of plane tilings

Part of: Designs and configurations Discrete geometry

Published online by Cambridge University Press:  09 April 2009

D. Coulson
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.

MSC classification

Secondary: 05B45: Tessellation and tiling problems 52C20: Tilings in $2$ dimensions 05B40: Packing and covering 52C15: Packing and covering in $2$ dimensions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 191 - 196
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Coulson, D., ‘A 15-colouring of 3-space omitting distance one’, Discrete Math. 256 (2002), 83–90.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), 194–196.CrossRefGoogle Scholar