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A coloring-book approach to finding coordination sequences.

C Goodman-Strauss1, N J A Sloane2

  • 1Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA.

Acta Crystallographica. Section A, Foundations and Advances
|December 22, 2018
PubMed
Summary
This summary is machine-generated.

A new graph coloring method simplifies finding coordination sequences for tilings. Surprisingly, Cairo tiling vertices share sequences with square tiling, proving a simple conjecture.

Keywords:
Cairo tilingcoordination sequencesdual tilingtetravalent verticestrivalent verticesuniform tiling

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Area of Science:

  • Graph theory
  • Discrete geometry
  • Combinatorics

Background:

  • Coordination sequences are crucial for understanding tiling structures.
  • Existing methods for determining these sequences can be complex.
  • The Cairo tiling (dual-3^2.4.3.4) and square tiling (4^4) are fundamental in geometric studies.

Purpose of the Study:

  • To introduce an elementary graph coloring method for deriving coordination sequences.
  • To investigate the coordination sequences of vertices in the Cairo tiling.
  • To apply the method to various uniform and Archimedean tilings.

Main Methods:

  • Graph coloring applied to the underlying graph of tilings.
  • Calculating coordination sequences for specific vertex types (tetravalent, trivalent).
  • Generalizing the method to diverse uniform tilings.

Main Results:

  • The coordination sequence for tetravalent vertices in the Cairo tiling is 1, 4, 8, 12, 16, ..., identical to that of the square tiling.
  • Coordination sequences were successfully determined for several uniform tilings (e.g., 3^2.4.3.4, 3.4.6.4, 4.8^2, 3.12^2, 3^4.6) and the snub-632 tiling.
  • The method provided proofs for previously conjectured formulas in several instances.

Conclusions:

  • The graph coloring approach offers a simplified and effective way to determine coordination sequences.
  • The surprising identity in coordination sequences between Cairo and square tilings is demonstrated.
  • This method advances the study of geometric structures and validates existing mathematical conjectures.