The Vertex Linear Arboricity of Integer Distance Graph G(Dm,1,4)

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in the induced subgraph of Vi(1 ≤ i ≤ k), where the vertex set Vi is the subset of vertices assigned color i. The vertex linear arboricity of a graphG is the minimum positive integer k such that G has a path k-vertex coloring. In this paper, we studied the vertex linear arboricity of the integer distance graphG (Dm,1,4), whereDm,1,4 = [1,m] \ [1, 4], and proved that vla (G (Dm,1,4)) = ⌈ m 7 ⌉ +1 for every integerm ≥ 6. Key–Words: Integer distance graph; Vertex linear arboricity; Path coloring