联图W_4+C_n的交叉数

On crossing numbers of join of W_4+C_n

联图G+H表示将G中每个点与H中的每个点连边得到的图。在Klesc M.给出联图W3+Cn的交叉数的基础上,应用反证法和排除法得到了联图W4+Cn的交叉数为Z(5,n) + n +|n/2|+ 4(n≥3)),并在Zarankiewicz猜想成立的前提下,根据证明,提出对Wm+Cn的交叉数的一个猜想:cr(Wm+Cn)=Z(m+1,n)+|m/2||m-1/2||n/2|+|m/2|+|n/2|+2,n≥3。其中Z(m,n)=|m/2||m-1/2||n/2||n-1/2|,m,n为非负整数。

By connecting each vertex of a graph G to each vertex of a graph H, a join graph, denoted by G＋H, is obtained. In this paper, based on the crossing number of W3 ＋ Cn obtained by Klesc M., it gets that the crossing number of W4 ＋ C is Z（5, n）＋ n ＋ [n/2] ＋ 4（n ≥ 3） by reduction to absurdity and elimination method, and gives a conjecture of the crossing number of W ＋ C within the conjecture of Zarankiewicz,cr（Wm＋Cm）=Z（m＋1,n）＋[m/2][m-1/2][n/2]＋[m/2]＋[n/2]＋2,n≥3.in which Z（m , n） =[m/2][m-1/2][n/2][n-1/2]、m,n is nonnegative integer.