Improved Bounds for Minimum Fault-Tolerant Gossip Graphs

A k-fault-tolerant gossip graph is a (multiple) graph whose edges are linearly ordered such that for any ordered pair of vertices u and v, there are k + 1 edge-disjoint ascending paths from u to v. Let τ(n, k) denote the minimum number of edges in a k-fault-tolerant gossip graph with n vertices. In this paper, we present upper and lower bounds on τ(n, k) which improve the previously known bounds. In particular, from our upper bounds, it follows that τ(n, k) ≤ nk 2 + O(n log n). Previously, it has been shown that this upper bound holds only for the case that n is a power of two.