On the crossing number of join product of the discrete graph with special graphs of order five - Michal Staš (Electronic Journal of Graph Theory and Applications (EJGTA), 2020)
The main aim of the paper is to give the crossing number of join product G+Dn for the disconnected graph G of order five consisting of the complete graph K4 and of one isolated vertex. In the proofs,  it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G onto the set of subgraphs which cross the edges of the graph G exactly one times. All methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn for two new graphs.