$K_{5}\times S_{n}$ 的交叉数

On the Crossing Numbers of $K_5\times S_n$

中文关键词:  图  交叉数  星  笛卡尔积.

英文关键词:graph  drawing  crossing number  star  Cartesian products.

中文摘要:

      把完全图$K_{5}$的五个顶点与另外$n$个顶点都联边得到一类特殊的图$H_{n}$.文中证明了$H_{n}$的交叉数为$Z(5,n)+2n+\lfloor \frac{n}{2}\rfloor+1$,并在此基础上证明了$K_{5}$与星$K_{1,n}$的笛卡尔积的交叉数为$Z(5,n)+5n+\lfloor\frac{n}{2} \rfloor+1$.

英文摘要:

      By connecting the $5$ vertices of $K_{5}$ to other $n$ vertices, we obtain a special family of graph denoted by $H_{n}$. This paper proves that the crossing number of $H_{n}$ is $Z(5,n)+2n+\lfloor \frac{n}{2} \rfloor+1$, and the crossing number of Cartesian products of $K_{5}$ with star $S_{n}$ is $Z(5,n)+5n+\lfloor \frac{n}{2} \rfloor+1$.