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

The crossing number 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 W₅ + Pn for the wheel W₅ on six vertices, where Pn is the path on n vertices. Staš and Valiska conjectured that the crossing number of Wm + Pn is equal to Z ( m + 1 ) Z ( n ) + ( Z ( m ) − 1 ) ⌊ n 2 ⌋ + n + 1 , for all m ≥ 3, n ≥ 2, where Zarankiewicz’s number is defined as Z ( n ) = ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ for n ≥ 1. Recently, this conjecture was proved for W₃ + Pn by Klešč and Schrötter, and for W₄ + Pn by Staš and Valiska. We establish the validity of this conjecture for W₅ + Pn . The conjecture also holds due to some isomorphisms for Wm + P₂, Wm + P₃ by Klešč, and for Wm + P₄ by Staš for all m ≥ 3.