On the Crossing Numbers of Join Products of Four Graphs of Order Six With the Discrete Graph

The aim of this paper is to extend known results concerning crossing numbers of graphs by giving the crossing number for join product G* + D-n of the connected graph G* of order six consisting of one 3-cycle and three leaves of which exactly two are adjacent with the same vertex of such 3-cycle, and D-n consists of n isolated vertices. The proofs rely on a partial classification of all subgraphs whose edges cross the edges of G* just once. Due to the mentioned algebraic topological approach, we extend known results concerning crossing numbers for join products of new graphs. Finally, by adding new edges to the graph G*, the crossing numbers of G(i) + D-n for three other graphs G(i) of order six will be also established.