The vertex linear arboricity of distance graphs

A distance graph is a graph G(R,D)G(R,D) with the set of all points of the real line as vertex set and two vertices u,v∈Ru,v∈R are adjacent if and only if |u-v|∈D|u-v|∈D where the distance set DD is a subset of the positive real numbers. Here, the vertex linear arboricity of G(R,D)G(R,D) is determined when D   is an interval between 1 and δδ. In particular, the vertex linear arboricity of integer distance graphs G(D)G(D) is discussed, too.