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Suppose that e is an edge in a weighted graph that is incident to a vertex v such that the weight of e does not exceed the weight of any other edge incident to v. Show that there exists a minimum spanning tree containing this edge.

Dhruv Pathak
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Proof by Contradiction

Assume that there exists a vertex v such that the MST does not use its smallest weight edge, e and instead uses another incident edge, let's call this x. Now, assume that we add the edge e back to the MST so that a cycle forms. We can now remove the previous edge used, x, on that cycle. At this point, we have another MST with a lower overall cost than the previously found spanning tree. This is a contradiction because the spanning tree with edge x was not actually an MST if it had a higher cost than the spanning tree with edge e.

anaidu
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