A Low-Diameter Nash-Williams--Tutte Theorem

The Nash-Williams--Tutte theorem characterizes the existence of $k$ edge-disjoint spanning trees by a condition on every vertex partition. We prove that, for simple graphs, the same condition guarantees $k$ edge-disjoint spanning trees, each of diameter $O(n/δ)$ and with $O((n/δ)\log(δ+1))$ internal vertices, where $n$ is the number of vertices and $δ$ is the minimum degree. Both bounds hold simultaneously for every tree, with absolute implicit constants independent of $k$, and each is asymptotically optimal in general. Our proof is constructive and yields a randomized algorithm with expected running time $\tilde{O}(n^5)$. We derive applications to short edge-disjoint routes and weak linkages, small Steiner trees, and survivable networks with short paths. The joint bounds also yield low-diameter spanning Eulerian subgraphs and short tours for postman and general routing problems.