Geodesic
Distance coloring of the hexagonal lattice
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 151-166

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Motivated by the frequency assignment problem we study the d-distant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted χ_d(H), is the minimum number of colors needed for a d-distant coloring of H. We give the exact value of χ_d(H) for any d odd and estimations for any d even.
Keywords: distance coloring, distant chromatic number, hexagonal lattice of the plane, hexagonal tiling, hexagonal grid, radio channel frequency assignment 
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Jacko, Peter; Jendrol', Stanislav. Distance coloring of the hexagonal lattice. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 151-166. http://geodesic.mathdoc.fr/item/DMGT_2005_25_1-2_a15/