In this paper, we show that the toroidal crossing number of K-4,K-n, K-1.3,K-n, K-2.2,K-n, K-1.1.2,K-n and K-1.1,K-1,K-1,K-n is left perpendicular n/4 right perpendicular vertical bar 2n - 4(1 + left perpendicular n/4 right perpendicular)vertical bar - In addition, a new lower bound for the toroidal crossing number of K. has been obtained. We also discuss about the crossing number of K-4,K-n in the general surfaces.
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