2-Distance Colorings of Integer Distance Graphs

Brahim Benmedjdoub (1) , Eric Sopena (2) , Isma Bouchemakh (1)

1 L'IFORCE - L'IFORCE
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 χ 2 (G) of G is then the mallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = {d 1 ,. .. , d k }, 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 χ 2 (G(D)) = ∆(G(D)) + 1.

Mots clés: 2-distance coloring; Integer distance graph.

Domaines: Mathématique discrète [cs.DM]

Brahim Benmedjdoub, Eric Sopena, Isma Bouchemakh. 2-Distance Colorings of Integer Distance Graphs. Discussiones Mathematicae Graph Theory, 2019, 39, pp.589-603. ⟨hal-01279943⟩