The crossing numbers of join product of four graphs on six vertices with discrete graphs

The main aim of the paper is to give the crossing number of the join product G* + D_n for the graph G* isomorphic to 4-regular graph on six vertices except for two distinct edges with no common vertex such that two remaining vertices are still adjacent, and where D_n consists of n isolated vertices. The proofs are done with possibility of an existence of a separating cycle in some particular drawing of the investigated graph G* and also with the help of well-known exact values for crossing numbers of join products of two subgraphs H_k of G* with discrete graphs.