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# Three-Colorable Map

A cubic map is three-colorable iff each interior region is bounded by an even number of regions. A non-cubic map bounded by an even number of regions is not necessarily three-colorable, as evidenced by the tetragonal trapezohedron (dual of the square antiprism), whose faces are all bounded by four other faces but which is not three-colorable (it has chromatic number 4). The Penrose tiles are known to be three-colorable (Babilon 2001).

In general polyform packing problems, the most elegant solutions are cubic and three-colorable. The illustration above shows a three-colorable packing of the 63 unholey (out of 64 total) double-L tetrominoes into a rectangle

Map Coloring, Four-Color Theorem, Tetragonal Trapezohedron, Three-Colorable Graph, Three-Colorable Knot

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## References

Babilon, R. "3-Colourability of Penrose Kite-and-Dart Tilings." Disc. Math. 235, 137-143, 2001.

## Referenced on Wolfram|Alpha

Three-Colorable Map

## Cite this as:

Weisstein, Eric W. "Three-Colorable Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Three-ColorableMap.html