Two floor building needing eight colors

Stéphane Bessy 1 Daniel Gonçalves 1 Jean-Sébastien Sereni 2
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of 3-dimensional parallelepiped arrangements. 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 colours. 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 or volume).
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Stéphane Bessy, Daniel Gonçalves, Jean-Sébastien Sereni. Two floor building needing eight colors. Journal of Graph Algorithms and Applications, Brown University, 2015, 19 (1), pp.1--9. ⟨hal-00996709v2⟩

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