整数距离图G(D_(m,k,2))的点荫度

图G的点荫度va(G)是顶点集合V(G)能划分成的这样一些子集的最少数目,其中任一子集的点导出子图都是森林.整数距离图G(D)以全体整数作为顶点集,顶点u,v相邻当且仅当|u-v|∈D,其中D是一个正整数集.对于m2k≥2,令D_(m,k,2)=1,m]\{k,2k}.该文得出了整数距离图G(D_(m,k,2))的点荫度的几个上、下界;进而,对于m≥4,有va(G(D_(m,1,2)))=(m+4)/5];对于m=10q+j,j=0,1,2,3,5,6,有va(G(D_(m,2,2)))=(m+1)/5]+1.

The Vertex Arboricity of Integer Distance Graphs G(Dm, k, 2)

The vertex arboricity va(G) of a graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces a subgraph whose connected components are trees. An integer distance graph is a graph G(D) with the set of all  integers as vertex set and two vertices u, v\in Z are adjacent if and only if |u-v in D where the  distance set D is a subset of the positive integers set. Let _{Dm, k, 2}=1,m]\k, 2k} for m>2k≥2. In this paper, some upper and lower bounds of the vertex arboricity of the integer distance graph G(D_{m, k, 2}) are obtained. Moreover, va(G(D_{m,1, 2}))=lm+4/5l  for m≥4 and  va(G(D_{m, 2, 2}))=lm+1/5l+1 for any positive integer m=10q+j with j =0,1, 2, 3, 5, 6.