Geodesic
Coloring of the dth Power of the Face-Centered Cubic Grid
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1001-1020

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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. We give lower and upper bounds on the chromatic number of the dth 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 the face-centered cubic grid.
Keywords: face-centered cubic grid, distance coloring, dth power of graph 
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Gastineau, Nicolas; Togni, Olivier. Coloring of the dth Power of the Face-Centered Cubic Grid. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1001-1020. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a8/