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Drawings of Cm×Cn with One Disjoint Family II

✍ Scribed by Hector A. Juarez; Gelasio Salazar

Book ID
102584407
Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
99 KB
Volume
82
Category
Article
ISSN
0095-8956

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✦ Synopsis

A long-standing conjecture states that the crossing number of the Cartesian product of cycles C m _C n 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 C m _C n 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 C m _C n is at least (m&2) nÂ2, for all integers m, n such that n m 4. This is the best general lower bound known for the crossing number of C m _C n .

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