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2 results

  • Decomposition of Multiple Packings with Subquadratic Union Complexity

  • JÁNOS PACH, BARTOSZ WALCZAK
  • Journal:
    Combinatorics, Probability and Computing / Volume 25 / Issue 1 / January 2016
    Published online by Cambridge University Press:
    04 January 2016, pp. 145-153

  • Suppose k is a positive integer and ${\cal X}$ is a k-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most k sets. Suppose there is a function f(n) = o(n2) with the property that any n members of ${\cal X}$ determine at most f(n) holes, which means that the complement of their union has at most f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that ${\cal X}$ can be decomposed into at most p (1-fold) packings, where p is a constant depending only on k and f.

  • On the chromatic number of plane tilings

  • Part of
  • D. Coulson
  • Journal:
    Journal of the Australian Mathematical Society / Volume 77 / Issue 2 / October 2004
    Published online by Cambridge University Press:
    09 April 2009, pp. 191-196
    Print publication:
    October 2004
    • Article
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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.