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The Diameter of the Minimum Spanning Tree of a Complete Graph

Abstract : Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].
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Louigi Addario-Berry, Nicolas Broutin, Bruce Reed. The Diameter of the Minimum Spanning Tree of a Complete Graph. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.237-248. ⟨hal-01184718⟩

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