Two floor building needing eight colors

TL;DR

This paper investigates the minimum number of colors needed to color adjacency graphs of 3D parallelepiped arrangements, motivated by frequency assignment problems, establishing bounds and examples for the chromatic number.

Contribution

It provides the first example of a 3D parallelepiped arrangement requiring exactly 8 colors and discusses bounds based on geometrical measures.

Findings

An arrangement within one floor requires exactly 8 colors.

The chromatic number can be bounded based on geometrical measures.

The adjacency graph's chromatic number can reach 8 in specific configurations.

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).