K_(1,1,m)□P_n的交叉数

On the crossing number of K_(1,1,m)□P_n

借助拉链积运算,Cartesian积图K(1,m)□Pn和K(2,m)□Pn的交叉数最近被先后确定.本文进一步证明了:对于m,n≧1,有cr(K(1,1,m)□Pn)=2n[m/2][(m-1)/2]+(n-1)[m/2].结论的证明基于Bokal关于树的Cartesian积图交叉数的有关结果.另外,我们也给出了确定K(2,m)□Pn交叉数的一个简洁方法.

The crossing numbers of the Cartesian products K1,m□Pn and K2,m□Pn were recently determined with the zip product operation. In this paper, we further show that cr（K（1,1,m）□Pn）=2n[m/2][（m-1）/2]＋（n-1）[m/2]for rt ≥ 1, m ≥1. Our proof depends on Bokal＇s results for Cartesian products with trees. Furthermore, we give a simple proof for the crossing number of K2,m□Pn.