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 Maderʹs Theorem, which bounds the number of edges in any edge-minimal k-connected graph.