The Crossing Number Of Flower Snark And K_m-e□P_n

Crossing number is an important concept measuring the non-planarity of graphs.Calculating the crossing number of a given graph is, in general, an elusive problem. In 1983,Garey and Johnson have proved that the problem of determining the crossing number of anarbitrary graph is NP-complete. A very few graphs' crossing number is determinied, such asthe generalized Petersen graph P (n,3), Cartesian products of paths with 5-vertex graphs. Thecrossing numbers of the complete graph and the complete bipartite graph are open problemsin topological graph theory.This paper focuses on the crossing number of the Flower Snark and related graphF_n (n≥3) and the Twisted Flower Snark and related graph F_n^* (n≥3). Both the exactvalues of the crossing number are determined. cr(F₃)=2,cr(F₄)=3,cr(F₅)=4,cr(F_n)=n(n≥6). cr(F₃^*)=1,cr(F₄^*)=2,cr(F₅^*)=4,cr(F_n^*)=n(n≥6).The crossing number of K_m-eâ–¡P_n is also studied in this paper. First, the upper boundof the crossing number of K_m-eâ–¡P_n is determined: cr(K_m-eâ–¡P_n)≤(n-1)(1／4[(m+2)／2][(m+1)／2][m／2][(m-1)／2]-[(m+1)／2][(m-1)／2]+[(m+1)／2]-[m／2])+1／2[(m+3)／2][m／2][(m-2)／2][(m-3)／2],m≥3,n≥1.At the same time, it is confirmed that the lower bound of the crossing numberof K_m-eâ–¡P_n is determined as follows: cr(K_m-eâ–¡P_n)≥(n-1)cr(K_m+2-2K₂)+2cr(K_m+1-e), m≥4, n≥1.Further more, it is proved that: cr(K₆-eâ–¡P_n)=12n, n≥1.