"On the crossing number of the join of the wheel on six vertices with a path"

Abstract

The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. The main aim of the paper is to give the crossing number of join product $W_5+P_n$ for the wheel $W_5$ on six vertices, where $P_n$ is the path on $n$ vertices. Sta\v s and Valiska conjectured that the crossing number of $W_m+P_n$ is equal to $Z(m+1)Z(n) + (Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n +1$, for all $m\geq 3$, $n\geq 2$, where Zarankiewicz's number is defined as $Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor $ for $n\geq 1$. Recently, this conjecture was proved for $W_3+P_n$ by Kle\v s\v c and Schr\"otter, and for $W_4+P_n$ by Sta\v s and Valiska. We establish the validity of this conjecture for $W_5+P_n$. The conjecture also holds due to some isomorphisms for $W_m+P_2$, $W_m+P_3$ by Kle\v s\v c, and for $W_m+P_4$ by Sta\v s for all $m\geq 3$.