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 $V_1, \ldots, V_k \subset V(G)$ where $V_i \neq \emptyset$ there are at least t(k − 1) crossing edges (Tutte 1961, Nash-Williams 1961)."For this article, we will 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.)

Related tree-packing properties
A k-arboric graph is necessarily k-edge connected. The converse is not true.

As a corollary of NW, every 2k-edge connected graph is k-arboric.

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

Nash-Williams theorem for forests
In 1964, Nash-Williams generalized the above result to forests:"U"A proof is given here.

This is how people usually define what it means for a graph to be t-aboric.

In other words, for every subgraph S = G[U], we have $$t \geq \lceil E(S) / (V(S) - 1) \rceil$$. It is tight in that there is a subgraph S that saturates the inequality (or else we can choose a smaller t). This leads to the following formula"$t = \lceil \max_{S \subset G} \frac{E(S)}{V(S) - 1} \rceil$"also referred to as the NW formula.

The general problem is to ask when a graph can be covered by edge-disjoint subgraphs.