Answer Happy

One of the basic motivations behind the Minimum Spanning Tree Problem is the goal of designing a spanning network for a set of nodes with minimum total cost. Here we explore another type of objective: designing a spanning network for which the most expensive edge is as cheap as possible. Specifically, let G = (V, E) be a connected graph with n vertices, m edges, and positive edge costs that you may assume are all distinct. Let T = (V, E') be a spanning tree of G; we define the bottleneck edge of T to be the edge of T with the greatest cost. A spanning tree T of G is a minimum bottleneck spanning tree if there is no spanning tree T' of G with a cheaper bottleneck edge. (a) Is every minimum-bottleneck tree of G a minimum spanning tree of G? Prove or give a counterexample. (b) Is every minimum spanning tree of G a minimum-bottleneck tree of G? Prove or give a counterexample.