Tutte's Theorem gives a precise combinatorial condition for a graph to have a perfect matching: a graph has a perfect matching if and only if, for every subset of its vertices, removing that subset leaves at most as many odd-sized connected components as the size of the removed subset. Proved by William Tutte, it generalizes Hall's Marriage Theorem from bipartite graphs to graphs in general.
Facts
StatementThe theorem states that a graph has a perfect matching if and only if, for every subset U of its vertices, removing U leaves the remaining graph with at most as many odd components as the size of U. 1 Classification
Statement Form Statement FormCharacterization Theorem 1 Connections
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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
Proved By
Source Tutte's Theorem (Graph Theory) (Wikipedia)
Sources
1. Tutte's Theorem (Graph Theory) (Wikipedia)
Wikimedia FoundationWikipedia, Tutte's theorem on perfect matchings, Tutte's theorem section
A graph, G = (V, E), has a perfect matching if and only if for every subset U of V, the subgraph G − U has at most |U| odd components (connected components having an odd number of vertices).
Proved By: W. T. Tutte, Lead paragraph
In the mathematical discipline of graph theory, the Tutte theorem, named after William Thomas Tutte, is a characterization of finite undirected graphs with perfect matchings. It
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