Nash-Williams theorem

In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:

A graph G has t edge-disjoint spanning trees iff for every partition ${\textstyle V_{1},\ldots ,V_{k}\subset V(G)}$ where ${\displaystyle V_{i}\neq \emptyset }$ there are at least t(k − 1) crossing edges.

The theorem was proved independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof.

For this article, we say that such a graph has arboricity t or is t-arboric. (The actual definition of arboricity is slightly different and applies to forests rather than trees.)