The Nash-Williams Formula gives the exact maximum number of edge-disjoint spanning trees a connected graph contains, expressing it as the minimum, taken over every partition of the vertex set into parts, of the number of edges crossing between parts divided by one less than the number of parts, rounded down. Named for Crispin Nash-Williams, who proved it independently alongside Bill Tutte, it is a foundational result of structural graph theory on how densely a graph's edges can be organized into disjoint spanning trees.
Facts
Classification
Statement Form Statement Form Statement FormCharacterization Theorem 1 StatementA graph G has t edge-disjoint spanning trees if and only if for every partition of the vertices into k nonempty parts there are at least t(k-1) crossing edges. 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Nash-Williams theorem (Wikipedia)
Proved By
Source Nash-Williams theorem (Wikipedia)
Sources
1. Nash-Williams theorem (Wikipedia)
Main Theorem section
A graph G has t edge-disjoint spanning trees iff for every partition V1, ..., Vk
- The theorem was proved independently by Tutte and Nash-Williams, both in 1961
In Branch: Graph Theory, Lead sentence
In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and mor
Proved By: W. T. Tutte, Lead paragraph
The theorem was proved independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof.
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