Conjectures about wheels in the join product with cycles

Abstract

By the crossing number $\mathrm{cr}(G)$ of a simple graph $G$ we understand the minimum number of edge crossings over all possible drawings of $G$ in the plane. The crossing numbers of the join product of two graphs have been studied for their wide application in practice. The main purpose of the paper is to establish $\mathrm{cr}\left(W_5+C_n\right)$ for the wheel $W_5$ on six vertices, where $C_n$ are the cycles on $n$ vertices. W. Yue et. al. in [Comp. Eng. Appl. 2014, 50 (18), 79-84] conjectured that the crossing number of $W_m+C_n$ is equal to $Z(m+1)Z(n)+\big(Z(m)-1\big)\big\lfloor \frac{n}{2} \big\rfloor+n+ \big\lceil\frac{m}{2}\big\rceil+2$ for all integers $m,n\geq 3$. Here, the Zarankiewicz's number $Z(n)=\big\lfloor \frac{n}{2} \big\rfloor \big\lfloor \frac{n-1}{2} \big\rfloor$ is defined for all $n\geq 1$. The mentioned conjecture was verified for $W_3+C_n$ by M. Klešč, and for $W_4+C_n$ by M. Staš and J. Valiska. We establish the validity of this conjecture for $W_5+C_n$.