Conjectures about wheels in the join product with cycles - Staš, M.; Timková, M. (Carpathian Mathematical Publications, 2025)
By the crossing number 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 cr(W_5+C_n) 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)+(Z(m)-1)n2 floor+n+ m2ceil+2 for all integers m,n≥ 3. Here, the Zarankiewicz's number Z(n)=n2 floor n-12 floor is defined for all n≥ 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.