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Keywords:

  • r-graphs;
  • generalized Berge–Fulkerson conjecture;
  • perfect matchings;
  • excessive index

Abstract

We construct a family of r-graphs having a minimum 1-factor cover of cardinality inline image (disproving a conjecture of Bonisoli and Cariolaro, Birkhäuser, Basel, 2007, 73–84). Furthermore, we show the equivalence between the statement that inline image is the best possible upper bound for the cardinality of a minimum 1-factor cover of an r-graph and the well-known generalized Berge–Fulkerson conjecture.