Deleted: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 <math display="inline">V_1, \ldots, V_k \subset V(G)</math> where <math>V_i \neq \emptyset</math> there are at least t(k&nbsp;−&nbsp;1) crossing edges (Tutte 1961, Nash-Williams 1961).

For this article, we will say that such a graph has arboricity&nbsp;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-aboric.

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

Nash-Williams theorem for forests

NW (1964) generalized the above result to forests:

G can be partitioned into t edge-disjoint forests iff for every <math>U \subset V(G)</math>, the induced subgraph G[U] has size <math>|G[U]| \leq t(|U|-1)</math>.

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 =&nbsp;G[U], we have <math>t \geq E(S) / (V(S) - 1)</math>. 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

<math>t = \max_{S \subset G} \frac{E(S)}{V(S) - 1}</math>

also referred to as the NW formula.

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

See also

• Arboricity
• Bridge (cut edge)
• Menger's theorem
• Tree packing conjecture