Geodesic
Two-floor buildings need eight colors
Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 1-9.

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Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of a 3-dimensional parallelepiped arrangement. In the case each parallelepiped is within one floor, a direct application of the Four-Colour Theorem yields that the adjacency graph has chromatic number at most 8. We provide an example of such an arrangement needing exactly 8 colors. We also discuss bounds on the chromatic number of the adjacency graph of general arrangements of 3-dimensional parallelepipeds according to geometrical measures of the parallelepipeds (side length, total surface area or volume).
DOI : 10.7155/jgaa.00344
Keywords: graph, coloring, contact graph, box 
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Stéphane Bessy; Daniel Gonçalves; Jean-Sébastien Sereni. Two-floor buildings need eight colors. Journal of Graph Algorithms and Applications, Tome 19 (2015) no. 1, pp. 1-9. doi : 10.7155/jgaa.00344. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00344/

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