Answer
Showing that if G is a weighted graph with distinct edge
weights, then for every simple circuit of G, the edge of
maximum weight in this circuit does not belong to any
minimum spanning tree of G.
Work Step by Step
-Suppose that a minimum spanning tree T contains edge e = uv that is the maximum weight edge in simple circuit C.
-Delete e from T .
- This creates a forest with two components, one containing u and the other containing v.
-Follow the edges of the path C − e, starting at u.
At some point this path must jump from the component of T − e containing u to the component of T − e containing v, say using edge f . This edge cannot be in T , because e can be the only edge of T joining
the two components (otherwise there would be a simple circuit
in T ).
- Because e is the edge of greatest weight in C, the weight of f is smaller.
-The tree formed by replacing e by f in T therefore has smaller weight, a contradiction.
