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I came across this problem in Introduction to algorithms Third Edition exercise.

So, how I proceeded was trying to contradict the situation when A minimum spanning tree has the largest edge greater than the largest edge of a bottleneck tree by cut and paste argument.

But it is not necessary that if I remove the largest Edge there exists an edge which connects this disconnected pieces into one!

Then, I tried proving it using Kruksal's Algorithm as basis as the maximum edge limits all other edges to be lesser than or equal to this maximum edge, but couldn't succeed.

Any help?

Shubham Sharma
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  • `But it is not necessary that if I remove the largest Edge there exists an edge which connects this disconnected pieces into one!` If you could remove the largest edge from a MST and replace it with a smaller one, that would mean the original wasn't a MST. – biziclop Oct 14 '15 at 15:33
  • What I was trying to prove with this thing was there exists no minimum spanning tree such that it is not bottleneck by assuming that there exists such a tree and contradicting it using cut and paste argument @biziclop – Shubham Sharma Oct 14 '15 at 15:37

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