Drawings of C-m x C-n with one disjoint family

We show that every drawing of C-m x C-n with either the m n-cycles pairwise disjoint or the n m-cycles pairwise disjoint has at least (m - 2)n crossin gs, for every m, n satisfying n greater than or equal to m greater than or equal to 3. This supports the long standing conjecture by Harary ct al. tha t the crossing number of C-m x C-n is (m - 2)n. (C) 1999 Academic Press.