论文标题
平面的存在$ 4 $连接基本上是$ 6 $ - 边缘连接的图形，没有爪子分解
The existence of planar $4$-connected essentially $6$-edge-connected graphs with no claw-decompositions
论文作者
论文摘要
在2006年，bar {á} t和托马森（Thomassen）猜想，每个平面$ 4 $ - 连接的$ 4 $ - 定型尺寸的简单图表可将三个分解的简单图形分解。后来，莱（Lai，2007年）被一个带有边缘连通性的平面图的家族反驳了这一猜想，最小的一个包含$ 24 $的顶点。在本说明中，我们首先给出一个较小的反例，只有$ 18 $的顶点，然后构建一个平面家族$ 4 $连接的本质上是$ 6 $ - 连接的$ 4 $ 4 $的尺寸尺寸的简单图形，可将三个尺寸除外，没有爪子分解。该结果为两个已知的结果提供了清晰度，这些结果表明，每三个$ 5 $连接的大小相连图的图形都可以承认，如果爪子分解本质上是$ 6 $ - 与边缘连接或平面。
In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known results which say that every $5$-edge-connected graph of size divisible by three admits a claw-decomposition if it is essentially $6$-edge-connected or planar.
