What is recorded here?
Tutte's1947 theorem characterizes when a finite graph can pair all its vertices: after deleting any vertex set S, the remaining number of odd-order connected components must be at most the size of S. A graph may represent which pairs of objects can be joined. Can every object be paired once without conflict? Each odd component left after removing S needs a different vertex from S to finish a pairing. Tutte proves that this necessary obstruction is also sufficient in the paper's finite graph class, using a Pfaffian encoding and structural arguments.
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Scientific scope
The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.
Read the original work
- W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile
- London Mathematical Society/Wiley metadata for Tutte's1947 Factorization of Linear Graphs
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People and systems
William T. Tutte
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English catalogue narrative
Tutte: a complete criterion for perfect matching in a finite graph
Tutte's1947 theorem characterizes when a finite graph can pair all its vertices: after deleting any vertex set S, the remaining number of odd-order connected components must be at most the size of S.
A graph may represent which pairs of objects can be joined. Can every object be paired once without conflict? Each odd component left after removing S needs a different vertex from S to finish a pairing. Tutte proves that this necessary obstruction is also sufficient in the paper's finite graph class, using a Pfaffian encoding and structural arguments.
Scientific scope
The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.
People and systems
William T. Tutte