嵌入曲面的图的点荫度

On the vertex-arboricity of embedded graphs

图G的导出森林k-划分是指其顶点集V(G)的一个k-划分(V1,V2,…,Vk),使得对于每个i(1≤i≤k),导出子图G[Vi]是一个森林。图G的点荫度是使得图G有导出森林k-划分的最小的正整数k,记为va(G)。主要证明了如果图G能够嵌入到欧拉示性数非负的曲面上,则当图G满足三类条件时,可以得到va(G)≤2。

An induced forest k-partition of a graph G is a k-partition（ V1,V2,…,Vk） of the vertex set V（ G） such that,for each i with 1≤i≤k,the induced subgraph G[Vi]is a forest. The vertex arboricity of a graph G is the minimum positive integer k such that G has an induced forest k-partition,denoted by va（ G）. Let G be a simple graph embedded in a surface of nonnegative Euler characteristic,and if G satisfies three kinds of conditions,then va（ G） ≤2.