Let $ G$ be a connected undirected graph, with integral positive weights on the edges, and let $ e_1$ be an edge of $ G$ . As part of an assignment I proved the following Lemma 1:
The edge $ e_1$ appears in any MST of $ G$ iff $ e_1$ is a light edge in any cut $ C$ that it crosses.
Is it equivalent to the following Lemma 2?
An edge $ 𝑒 = (𝑢, 𝑣)$ appears in all MSTs of $ 𝐺$ iff every cycle including $ 𝑒$ has an edge $ 𝑒′$ with $ 𝑤(𝑒) < 𝑤(𝑒’)$ .
The latest is clearly a negation of the red rule for MSTs. I think they are equivalent but can’t prove it.
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