最大度是4的可平面图是第一类图的充分条件

Some sufficient conditions for a planar graph of maximum degree four to be Class 1

运用Discharge方法证明:最大度是4,且满足下列条件之一的可平面图G是第一类的.(1)G中不含长度为4至9的圈;(2)G中不含4-圈和5-圈,且任意两个3-面不关联于同一个顶点;(3)G中不含长度在5和8之间的圈,且任意两个3-圈,任意两个4-圈不关联于同一个顶点;(4)围长不小于4,G中不含有弦的8-圈,且任意两个4-面不关联于同一个顶点.

By applying discharging method,we showed that a planar graph G with maximum degree four and girth g is of class 1,if it satisfies one of the following conditions. （1） G does not contain cycles of length from 4 to 9;（2） G does not contain 4-cycles and 5-cycles,and two 3-faces are not incident with a common vertex;（3） G does not contain cycles of length from 5 to 8,and two 3-cycles,two 4-cycles are not incident with a common vertex;（4） g≥4,G does not contain chordal-8-cycle,and two 4-faces are not incident with a common vertex.