Here are 3 exercises for Wednesday's recitation.  

(1) Let (u,v) be a minimum-weight edge in a graph $G$.  Show that
(u,v) belongs to some minimum spanning tree of G.

(2) Prove that the weight of the maximum-weight edge in a minimum
spanning tree is less than or equal to the weight of the
maximum-weight edge in any other spanning tree.

(3) Suppose that a graph G has a minimum spanning tree already
computed.  How quickly can you update the minimum spanning tree if the
weight of an edge changes?
