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: <blockquote>A graph G has t edge-disjoint spanning trees iff for every partition where there are at least t(k â 1) crossing edges. </blockquote> 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:<blockquote>A graph can be partitioned into edge-disjoint forests iff for every , the induced subgraph has at most edges. </blockquote>Other proofs are given here.
This is how people usually define what it means for a graph to be t-arboric.
In other words, for every subgraph , we have . It is tight in that there is a subgraph that saturates the inequality (or else we can choose a smaller ). This leads to the following formula<blockquote>,</blockquote>also referred to as the Nash-Williams formula.
The general problem is to ask when a graph can be covered by edge-disjoint subgraphs.