Two-floor buildings need eight colors

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). Submitted: May 2014 Reviewed: August 2014 Revised: September 2014 Accepted: December 2014 Final: January 2015 Published: January 2015 Article type: Regular paper Communicated by: J.S.B. Mitchell The second and third author’s work were partially supported by the French Agence Nationale de la Recherche under references ANR-12-JS02-002-01 and ANR-10-JCJC-0204-01, respectively. E-mail addresses: [email protected] (S. Bessy) [email protected] (D. Gonçalves) [email protected] (J.-S. Sereni) 2 Bessy, Gonçalves, and Sereni Two-floor buildings need eight colors