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

A long-standing conjecture states that the crossing number of the Cartesian product of cycles C-m x C-n is (m - 2) n, for every m, n satisfying n grea ter than or equal to m greater than or equal to 3. A crossing is proper if it occurs between edges in different principal cycles. In this paper drawin gs of C-m x C-n with the principal n-cycles pairwise disjoint or the princi pal ill-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 C-m x C-n is at least (m - 2) n/2, for all integers m, n such that n greater than or equal to m grea ter than or equal to 4. This is the best general lower bound known for the crossing number of C-m x C-n. (C) 2001 Academic Press.