On the crossing numbers of K m ⊔C n and K m,l ⊔P n

Ringeisen and Beineke have proved that cr(C"[email protected]?C"n)=n and cr(K"[email protected]?C"n)=3n. Bokal has proved that cr(K"1","[email protected]?P"n)=(n-1)@[email protected][email protected][email protected]?. In this paper we study the crossing numbers of K"[email protected]?C"n and K"m","[email protected]?P"n, and show (i) cr(K"[email protected]?C"n)>=n.cr(K"m"+"2) for n>=3 and m>=5; (ii) cr(K"[email protected]?C"n)==8 with even n>=4, and equality holds for m=5,6,7 and for m=8,9,10 with even n>=4 and (iii) cr(K"m","[email protected]?P"n)= =2, and equality holds for min(m,l)=2.