A Coloring Book Approach to Finding Coordination Sequences

TL;DR

This paper introduces a simple coloring-based method to determine coordination sequences in tilings, providing proofs for some previously conjectured formulas and applying it to various uniform tilings.

Contribution

The paper presents a novel elementary coloring approach for finding coordination sequences in tilings, simplifying proofs and extending results to multiple uniform tilings.

Findings

Coordination sequence for tetravalent vertices is 1, 4, 8, 12, 16, ...

Method applies successfully to various uniform tilings

Provides proofs for some previously conjectured formulas

Abstract

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. We illustrate the method by first applying it to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3^2.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the same as for a vertex in the familiar square (or 4^4) tiling. We thought that such a simple fact should have a simple proof, and this article is the result. We also apply the method to obtain coordination sequences for the 3^2.4.3.4, 3.4.6.4, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and bew tilings. In several cases the results provide proofs for previously conjectured formulas.