Nash-Williams theorem

Template:Short description 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 $V_1,\dots ,V_k\subset V(G)$ where $V_i\ne \mathrm{\varnothing }$ 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.)

Related tree-packing properties

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.

Nash-Williams theorem for forests

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

A graph $G$ can be partitioned into $t$ edge-disjoint forests iff for every $U\subset V(G)$, the induced subgraph $G[U]$ has at most $t(|U|-1)$ edges.

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 $S=G[U]$, we have $t\ge \lceil E(S)/(V(S)-1)\rceil$. 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

$t=\lceil \underset{S\subset G}{max}\frac{E(S)}{V(S)-1}\rceil$,

also referred to as the Nash-Williams formula.

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

See also

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

External links

• Paulson, Lawrence C. The Nash-Williams partition theorem (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)