论文标题
在没有4个周期和6个周期的森林和路径不相交的情况下，将平面图和6个循环分配
Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths
论文作者
论文摘要
在本文中，我们表明，每个平面图没有$ 4 $ - 周期和6美元的周期，其顶点将其分为两套，其中一组会引起森林，而另一组则最大程度为$ 2 $（相当于，与路径的不相交联盟）。 请注意，我们可以将森林的顶点集分为两个独立的集合。但是，一对独立组合可能不会引起森林。因此，我们的结果扩展了Wang and Xu（2013）的结果，指出每个平面图的顶点集，而无需$ 4 $ -CYCLES和$ 6 $ -CYCLES可以分为三组，其中一组诱导了最高二级图，其余两个是独立的。
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, where one induces a graph with maximum degree two, and the remaining two are independent sets.
