On the Crossing Number of a Torus Network

Reducing the number of link crossings in a network drawn on plane such as wiring board is well-known problem, and especially calculation minimum crossings: this crossing problem. It has been shown that finding general solution to problem NP-hard. So, addressed for particular classes graphs also our approach paper. More precisely, we focus hereinafter torus topology. First, discuss an upper bound cr(T(2, k)) 2-dimensional k-ary T(2, k) where k ≥ 2: result ≤ k(k - 2) given constructive proof lay foundations rest Second, extend discussion derive 3-dimensional torus: cr(T(3, 2k4 k3 4k2 2⌈k/2⌉⌊k/2⌋(k (k mod 2)) obtained. Third, n-dimensional derived from previously established results, with order additionally more clarity: cr(T(n, O(n2k2n-2) when n O(nk2n-1) otherwise.