---
tag: BYJ5EZMP6W
kind: statement
role: proposition
name: Nash-Williams theorem
sources:
  - wikipedia: en:60654680@r1332584259
    locator: Nash-Williams theorem
    note: adopted
    mention: wikipedia://en/60654680
    sections: [Statement]
same_as:
  wikidata: Q65066994
  wikipedia: Nash-Williams theorem
inventory: [PR9BT1BA40]
---

## Statement

In graph theory, the **Nash-Williams theorem** is a **tree-packing** theorem that describes how many edge-disjoint [[spanning trees|0FQ5B8S1ZC]] (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.

The theorem was proved independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof.