# 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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Chassaing, Philippe and others. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.237-248, 2006, DMTCS Proceedings
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https://hal.inria.fr/hal-01184718
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• HAL Id : hal-01184718, version 1

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Louigi Addario-Berry, Nicolas Broutin, Bruce Reed. The Diameter of the Minimum Spanning Tree of a Complete Graph. Chassaing, Philippe and others. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.237-248, 2006, DMTCS Proceedings. 〈hal-01184718〉

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