On the Crossing Numbers of $K_5\times S_n$

Abstract:

      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 $Z(5,n)+5n+\lfloor \frac{n}{2} \rfloor+1$.