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

Motivated by the problem of determining the crossing number of the Cartesian product C-m x C-n of two cycles, we introduce the notion of an (m,n)-arrangement, which is a generalization of a planar drawing of Pn+1 x C-m 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 C-7 x C-n is 5n, in agreement with the general conjecture that the crossing number of C-m x C-n is (m - 2)n, for 3 less than or equal to m less than or equal to n. (C) 2003 Elsevier Inc. All rights reserved.