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

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.