平面图点荫度的一个局部条件  

A Local Condition on the Vertex-arboricity of Planar Graphs

作　　者：黄丹君 凌银 HUANG Danjun;LING Yin(College of Mathematics and Computer Science,Zhejiang Normal University,Jinhua,Zhejiang,321004,P.R.China)

机构地区：[1]浙江师范大学数学与计算机科学学院,浙江金华321004

出　　处：《数学进展》2020年第2期146-158,共13页Advances in Mathematics（China）

基　　金：浙江省自然科学基金(No.LY18A010014)。

摘　　要：图G的点荫度va(G)是指V(G)的最小划分数,使得每个点划分集的导出子图是一个森林.众所周知,平面图G的点荫度至多为3.2008年,Raspaud和王维凡证明了:若G是不含κ-圈(κ∈{3,4,5,6})的平面图,则va(G)≤2.本文推广了上述结果,得到了点荫度至多为2的一个局部条件,即若平面图G的每一个点都不同时与3-圈、4-圈、5-圈和6-圈关联,则va(G)≤2.The vertex-arboricity va(G) of a graph G is the minimum number of partitions of vertices,so that each partition induces a forest.It is well known that va(G)≤3 for any planar graph G.In 2008,Raspaud and Wang proved that if G is a planar graph and does not containκ-cycle,κ∈{3,4,5,6},then va(G)≤2.In this paper,we obtain a new local condition.That is:if any vertex of planar graph G is not simultaneously incident with 3-cycle,4-cycle,5-cycle and 6-cycle,then va(G)≤2.

关 键 词：平面度 点荫度 染色 圈 

分 类 号：O157.5[理学—数学]