A tight bound on the collection of edges in MSTs of induced subgraphs

Let G = (V, E) be a complete n-vertex graph with distinct positive edge weights. We prove that for k ∈ {1, 2, ..., n - 1}, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of G with n - k + 1 vertices has at most n k - ((k + 1; 2)) elements. This proves a conjecture of Goemans and Vondrák [M.X. Goemans, J. Vondrák, Covering minimum spanning trees of random subgraphs, Random Structures Algorithms 29 (3) (2005) 257-276]. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal k-connected graph. © 2008 Elsevier Inc. All rights reserved.