Drawings of Cm×Cn with One Disjoint Family II

A long-standing conjecture states that the crossing number of the Cartesian product of cycles Cm×Cn is (m−2) n, for every m, n satisfying n⩾m⩾3. A crossing is proper if it occurs between edges in different principal cycles. In this paper drawings of Cm×Cn with the principal n-cycles pairwise disjoint or the principal m-cycles pairwise disjoint are analyzed, and it is proved that every such drawing has at least (m−2) n proper crossings. As an application of this result, we prove that the crossing number of Cm×Cn is at least (m−2) n/2, for all integers m, n such that n⩾m⩾4. This is the best general lower bound known for the crossing number of Cm×Cn.