## Abstract

Say that a graph G has property (Formula presented.) if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set (Formula presented.) and let e_{1}, e_{2}, … e_{N} be a uniformly random ordering of the edges of K_{n}, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge e_{m + 1} exactly if Gm ∪ {e_{m + 1}} has property (Formula presented.). We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?.

Original language | English |
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Pages (from-to) | 1272-1302 |

Number of pages | 31 |

Journal | Random Structures and Algorithms |

Volume | 57 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2020 |

## Keywords

- matching
- perfect matching
- random graph
- random process
- vertex cover