König-Egerváry Graphs are Non-Edmonds

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König-Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125-130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors ∈ ℝE satisfying two families of constraints: 'vertex saturation' and 'blossom'. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs-one combinatorial, one algorithmic-of its title's assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. (J Combin Theory Ser B 92:319-324, 2004).

Original languageEnglish
Pages (from-to)721-726
Number of pages6
JournalGraphs and Combinatorics
Issue number5
StatePublished - Sep 2010


  • Covering
  • Matching
  • Perfect matching polytope


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