Arrangements, circular arrangements and the crossing number of C7× Cn

Authors:
Jay Adamsson;R. Bruce Richter
Affiliations:
Department of Mathematics and Statistics, University of Prince Edward Island, Charlottetown PEI, Canada C1A 4P3;Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. West, Waterloo, Ont., Canada N2L 3G1
Venue:
Journal of Combinatorial Theory Series B
Year:
2004

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Abstract

Motivated by the problem of determining the crossing number of the Cartesian product Cm × Cn of two cycles, we introduce the notion of an (m, n)-arrangement, which is a generalization of a planar drawing of Pn+1 × Cm in which the two "end cycles" are in the same face of the remaining n cycles. The main result is that every (m, n)-arrangement has at least (m - 2)n crossings. This is used to show that the crossing number of C7 × Cn is 5n, in agreement with the general conjecture that the crossing number of Cm × Cn is (m - 2)n, for 3 ≤ m ≤ n.