A coloring-book approach to finding coordination sequences

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3².4.3.4) 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 4⁴) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 3².4.3.4, 3.4.6.4, 4.8², 3.12² and 3⁴.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.