# The Longest Minimum-Weight Path in a Complete Graph

@article{AddarioBerry2009TheLM, title={The Longest Minimum-Weight Path in a Complete Graph}, author={Louigi Addario-Berry and Nicolas Broutin and G{\'a}bor Lugosi}, journal={Combinatorics, Probability and Computing}, year={2009}, volume={19}, pages={1 - 19} }

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8].

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