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. © 1999 Academic Press.

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