Authors: B. Benmedjdoub, I. Bouchemakh, É. Sopena Title: 2-distance colorings of integer distance graphs Source: Discussiones Mathematicae Graph Theory Received 05.12.2016, Revised 12.11.2017, Accepted 13.11.2017, doi: 10.7151/dmgt.2040 | |
Abstract: 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 smallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D={d_{1},...,d_{ℓ}}, 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))=Δ(G(D))+1. | |
Keywords: 2-distance coloring, integer distance graph | |
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