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Let $G=(V,E)$ be a weighted undirected connected graph and $w: E \to \mathbb{R^{+}}$ a weight function so that there are no two edges that have the same weight, and $T$ is an MST of $G$ . Then in each cycle in the graph, the edge with the minimum weight belongs to $T$.

I either need to prove it in a positive way or give a counterexample.

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    – D.W.
    Feb 3, 2021 at 19:46

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Following is a simple counterexample: Take a complete graph on four vertices: ${u_{1},u_{2},u_{3},u_{4}}$, with edge weights $w(u_{1},u_{2}) = 1$, $w(u_{2},u_{3}) = 2$, $w(u_{1},u_{3}) = 4$, $w(u_{1},u_{4}) = 5$, $w(u_{2},u_{4}) = 3$, and $w(u_{3},u_{4}) = 6$. Here MST is composed of edges: $(u_{1},u_{2})$, $(u_{2},u_{3})$, and $(u_{2},u_{4})$ with total weight $6$. And, no edge of the cycle $(u_{1},u_{3},u_{4})$ is part of the MST.

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    $\begingroup$ I'm sure there are pliny others too. $\endgroup$
    – Jasen
    Feb 3, 2021 at 10:52
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    $\begingroup$ Kudos for the simplest possible example. $\endgroup$
    – vonbrand
    Feb 6, 2021 at 17:06

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