The crossing numbers of join products of seven graphs of order six with paths and cycles

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 this paper is to give the crossing numbers of the join products of seven graphs on six vertices with paths and cycles on n vertices. The proofs are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs. Finally, for m at least three and n = 5, we also establish the validity of a conjecture introduced by Stas ˇand Valiska concerning the crossings numbers of the join products of the wheels on m + 1 vertices with the paths on n vertices.