Geodesic
2-Distance Colorings of Integer Distance Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 589-603

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A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors 1,. . ., k such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number χ2(G) of G is then the smallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = d1, . . ., dℓ, the integer distance graph G = G(D) is the infinite graph defined by V (G) = ℤ and uv ∈ E(G) if and only if |v − u| ∈ D. We study the 2-distance chromatic number of integer distance graphs for several types of sets D. In each case, we provide exact values or upper bounds on this parameter and characterize those graphs G(D) with χ2(G(D)) = ∆(G(D)) + 1.
Keywords: 2-distance coloring, integer distance graph 
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     title = {2-Distance {Colorings} of {Integer} {Distance} {Graphs}},
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Benmedjdoub, Brahim; Bouchemakh, Isma; Sopena, Éric. 2-Distance Colorings of Integer Distance Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 589-603. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a20/