中文摘要：

图G的点荫度a（G）是G的使得每个子集诱导一个森林的顶点划分中子集的最少个数.我们熟知对任何平面图G,a（G）≤3,且对任何直径最大是2的平面图有a（G）≤2.文献[European J.Combin.,2008,29（4）：1064-1075]中给出下列猜想：任何没有3-圈的平面图都有一个顶点的划分（V1,V2）使得V1是独立集,V2诱导一个森林.本文证明了任何2-边连通上可嵌入的3-正则图G（G≠K4）都有一个顶点的划分（V1,V2）使得V1是独立集,V2诱导一个森林.

英文摘要：

The vertex-arboricity a（G） of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well known that a（G） ≤ 3 for any planar graph G, and that a（G） ≤ 2 for any planar graph G of diameter at most 2. The conjecture that every planar graph G without 3-cycles has a vertex partition （V1, V2） such that V1 is an independent set and V2 induces a forest was given in [European J. Corabin., 2008, 29（4）： 1064-1075]. In this paper, we prove that a 2-edge-connected cubic graph which satisfies some condition has this partition. As a corollary, we get the result that every up-embeddable 2-edge-connected cubic graph G（G≠K4） has a vertex partition （V1, V2） such that V1 is an independent set and V2 induces a forest.