The Crossing Numbers Of Joint Of Two Special Graphs

The crossing number of graph is mainly concerned with how to draw a picture on plane so that the number of crossing between edges is minimum. The crossing number cr（G） of graph G is the minimum number of the crossings between edges over all drawing of G on plane. In recent years,many researchers had studied the crossing numbers of the joint of graphs with order 5 or 6 with n isolated vertices. However,there are few results about the crossing numbers of Joint of the Graph on n（≥ 7） vertices with n isolated vertices. In this paper, by using the good drawing of the subgraph’s separating cycles,the crossing number of the joint graph on two special graphs with n isolated vertices was proved . The main contents include:（1）determined cr(G_m^（1）+K₂) and obtained the crossing number between G_m^（1） and Tⁱ when Tⁱ located in different region G_m^（1） of separating cycles. Based on above the results, by using with the mathematical induction and the reduction to absurdity, we proved the crossing number of the graph G_m^（1）+Kn.（2）we first determined cr(G_m^（2）+K₁)and cr(G_m^（2）＋K₂), obtained the crossing number between G_m^（2） and T’ when Tⁱ located in different region G_m^（2） of separating cycles. Based on above the results,by using the mathematical induction and the reduction to absurdity, we proved the crossing number of the graph G_m^（2）+K_n.