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Unlike the sum in the usual shortest path problem, the bottleneck cost of a nonempty graph path is its largest edge weight. The optimum is the smallest threshold at which the endpoints become connected by edges of weight at most that threshold. For a finite undirected graph, sorting edges and increasing the threshold, with a disjoint-set data structure maintaining components, finds this value. In the same undirected setting, the minimax path property of a minimum spanning tree provides a single tree representing such optimal routes for every pair of vertices.

Take a minimum spanning tree $T$, for example one constructed by Kruskal's algorithm. Fix $v$ and let $e$ be a longest edge of the tree's unique $xv$ path. Deleting $e$ partitions $T$ into two components separating $x$ and $v$. Any alternative $xv$ path crosses this graph cut. If all of its edges were shorter than $e$, a crossing edge $f$ would let us replace $e$ by $f$, producing a spanning tree of smaller total length. This contradicts minimality of $T$.

Therefore every competing path has an edge at least as long as $e$, and the tree path solves the bottleneck shortest path problem. The same minimum spanning tree works simultaneously for every $v$; this is the minimax path property of a minimum spanning tree.