Chromatic numbers for contact graphs of congruent cuboids
Søren Eilers, Rune Johansen, Rasmus Veber Rasmussen, Carsten Thomassen
TL;DR
The paper investigates chromatic numbers of contact graphs formed by configurations of mutually congruent cuboids with integer dimensions in 3D. It develops a framework combining rescaling, partitioning, degeneracy bounds, periodic colorings, and computer-assisted searches to obtain universal bounds (χ1 ≤ 8; with rotations ≤48) and to compute or bound values for specific dimensions, frequently finding χ up to 6 while many cases remain open. Exact values are established for several small cuboid shapes, and extensive pseudorandom searches yield strong lower bounds and illustrative configurations, though large gaps between upper and lower bounds persist. The work blends geometric constructions, graph-theoretic bounds, and computational experimentation, offering insights relevant to LEGO-style contact graphs and guiding future investigations in higher dimensions.
Abstract
We initiate the study of chromatic numbers for contact graphs of configurations of integer-sized cuboids in three dimensions, all of which are mutually congruent. Disallowing rotations, we show a global upper bound of 8 for the chromatic numbers, which implies that there is a global upper bound of 48 when the cuboids may be rotated freely. Specializing further to cuboids that are required to have a side length of one we obtain more precise upper bounds. Such upper bounds are compared to examples of configurations having relatively large chromatic numbers, leading to a complete determination of some of these chromatic numbers, but in general, the gaps between our upper and lower bounds are rather wide. In particular, we know of no such configuration of any size leading to a chromatic number above 6.