Coloring of the dth power of the face-centered cubic grid

TL;DR

This paper investigates the coloring properties of the face-centered cubic grid's d-th power, providing bounds on the chromatic number and exact values for specific cases, which aids in understanding sphere packing and contact graphs.

Contribution

It establishes new bounds on the chromatic number of the d-th power of the face-centered cubic grid, including exact results for d=2 and improved bounds for d=3.

Findings

Chromatic number of the grid for d=2 is exactly 13.

Provided bounds for the chromatic number for general d.

Sharper bounds for subgraphs of the face-centered cubic grid.

Abstract

The face-centered cubic grid is a three dimensional 12-regular infinite grid. This graph represents an optimal way to pack spheres in the three-dimensional space. In this grid, the vertices represent the spheres and the edges represent the contact between spheres. We give lower and upper bounds on the chromatic number of the d th power of the face-centered cubic grid. In particular, in the case d = 2 we prove that the chromatic number of this grid is 13. We also determine sharper bounds for d = 3 and for subgraphs of of the face-centered cubic grid.