Vertex arboricity of integer distance graph

Let D be a subset of the positive integers. The distance graph G ( Z , D ) has all integers as its vertices and two vertices x and y are adjacent if and only if | x − y | ∈ D , where the set D is called distance set. The vertex arboricity v a ( G ) of a graph G is the minimum number of subsets into which vertex set V ( G ) can be partitioned so that each subset induces an acyclic subgraph. In this paper, the vertex arboricity of graphs G ( Z , D m , k ) are studied, where D m , k = { 1 , 2 , … , m } ∖ { k } . In particular, v a ( G ( D m , 1 ) ) = ⌈ m + 3 4 ⌉ for any integer m ≥ 5 ; v a ( G ( D m , 2 ) ) = ⌈ m + 1 4 ⌉ + 1 for m = 8 l + j ≥ 6 and j ≠ 7 , and ⌈ m 4 ⌉ + 1 ≤ v a ( G ( D m , 2 ) ) ≤ ⌈ m 4 ⌉ + 2 for m = 8 l + 7 .