On the total_k-diameter of connection networks

We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of total_k-distance for a pair of nodes and total_k-diameter of a connection network, and study the value TD_k(d) which is the maximal such total_k-diameter of a network with diameter d. These notions have applications in fault-tolerant routing problems, in ATM networks, and in compact routing in networks. We prove an upper bound on TD_k(d) and a lower bound on the growth of TD_k(d) as functions of k and d; those bounds are tight, θ(d^k), when k is fixed. Specifically, we prove that TD_k(d)≤2^k-1d^k, with the exceptions TD₂(1) = 3, TD₃(1) = 5, and that for every k, d₀>0, there exists (a) an integer d≥d₀ such that TD_k(d)≥d^k/k^k, and (b) a k-connected simple graph G with diameter d such that d≥d₀, and td_k(G)≥(d-2)^k/k^k.