The toroidal crossing number of

In this paper, we show that the toroidal crossing number of K 4 , n , K 1 , 3 , n , K 2 , 2 , n , K 1 , 1 , 2 , n and K 1 , 1 , 1 , 1 , n is ⌊ n 4 ⌋ [ 2 n − 4 ( 1 + ⌊ n 4 ⌋ ) ] . In addition, a new lower bound for the toroidal crossing number of K m , n has been obtained. We also discuss about the crossing number of K 4 , n in the general surfaces.