Drawings of Cm×Cn with One Disjoint Family

We show that every drawing of Cm×Cn with either the m n-cycles pairwise disjoint or the n m-cycles pairwise disjoint has at least (m−2) n crossings, for every m, n satisfying n⩾m⩾3. This supports the long standing conjecture by Harary et al. that the crossing number of Cm×Cn is (m−2) n.