5. A spanning tree T is a bottleneck spanning tree if the maximum edge cost in T is as small as possible from among all spanning trees. a Show that a minimum spanning tree of G is also a bottleneck spanning tree. (Hint: Let T be a minimum spanning tree and (u, v) be the edge of T with maximum weight. If T is not a bottleneck spanning tree, let T" be the bottleneck spanning tree, then every edge in T" has weight less than (u, v). Find the contradiction. You may want to use the property we discussed in class (the second bullet of page 14 in slide 7).) b Show that a bottleneck spanning tree of G is not necessarily a minimum spanning tree. (By a counterexample) 5. A spanning tree T is a bottleneck spanning tree if the maximum edge cost in T is as small as possible from among all spanning trees. a Show that a minimum spanning tree of G is also a bottleneck spanning tree. (Hint: Let T be a minimum spanning tree and (u, v) be the edge of T with maximum weight. If T is not a bottleneck spanning tree, let T" be the bottleneck spanning tree, then every edge in T" has weight less than (u, v). Find the contradiction. You may want to use the property we discussed in class (the second bullet of page 14 in slide 7).) b Show that a bottleneck spanning tree of G is not necessarily a minimum spanning tree. (By a counterexample)