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.)

A k-arboric graph is necessarily k-edge connected. The converse is not true.

As a corollary of the Nash-Williams theorem, every 2k-edge connected graph is k-arboric.

Both Nash-Williams' theorem and Menger's theorem characterize when a graph has k edge-disjoint paths between two vertices.

In 1964, Nash-Williams generalized the above result to forests: