整数距离图G(D_(m，k，2))的点线性荫度

THE VERTEX LINEAR ARBORICITY OF THE INTEGER DISTANCE GRAPH G(D_(m,k,2))

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．本文讨论整数距离图的点线性荫度,记为vla(G(D))．对于m≥5k,设D_(m,k,2)={1,2,…,m}/{k,2k),得到vla(G(D_(m,1,2)))=■并决定出了G(D_(m,2,2))在某些特殊的仇值上点线性荫度的确切值以及当k≥3时G(D_(m,k,2))的点线性荫度的上、下界．

An integer distance graph is a graph G（D） with the set of all integers Z as vertex set and two vertices u, v E Z are adjacent if and only if ｜u - v｜ E D, where the distance set D is a subset of positive integers. Here the vertex linear arboricity of integer distance graph G（D） （denoted by vla（G（D））） is studied. Let Din,k,2 = {1, 2,……, m}／{k, 2k} for m ≥ 3k. In this paper, it is obtained that vla（G（Dm,1,2））=[m/5]＋1,[m＋1/5]＋1≤vla（G（Dm,2,2））≤{2[m/10],ifm=10l＋1,2[m/10]＋1,ifm=10l＋j,2≤j≤4,2[m/10]＋1,else. The exact values of the vertex linear arboricity of G（Dm,2,2） for some special m, and the upper and lower bounds of the vertex linear aboricity of G（Dm,k,2） for k≥ 3 are obtained.