## Abstract

Let G be a simple graph with vertex set V(G). A set S⊆V(G) is independent if no two vertices from S are adjacent. The graph G is known to be König–Egerváry if α(G)+μ(G)=|V(G)|, where α(G) denotes the size of a maximum independent set and μ(G) is the cardinality of a maximum matching. A nonempty collection Γ of maximum independent sets is König–Egerváry if |⋃Γ|+|⋂Γ|=2α(G) (Jarden et al., 2015). In this paper, we prove that G is a König–Egerváry graph if and only if for every two maximum independent sets S_{1},S_{2} of G, there is a matching from V(G)−S_{1}∪S_{2} into S_{1}∩S_{2}. Moreover, the same is true, when instead of two sets S_{1} and S_{2} we consider an arbitrary König–Egerváry collection.

Original language | English |
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Pages (from-to) | 175-180 |

Number of pages | 6 |

Journal | Discrete Applied Mathematics |

Volume | 231 |

DOIs | |

State | Published - 20 Nov 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2016 Elsevier B.V.

## Keywords

- Core
- Corona
- König–Egerváry collection
- König–Egerváry graph
- Maximum independent set
- Maximum matching