## Abstract

We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced in [3]. The process starts with an empty graph on the vertex set [n]. For every positive integers q and 1 ≤ r≤ n, in the ((q- 1 ) n+ r) th round of the process, the decision-maker, called Builder, is offered the vertex π_{q}(r), where π_{1}, π_{2}, … is a sequence of permutations in S_{n}, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to π_{q}(r). For several natural graph properties, such as k-connectivity, minimum degree at least k, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze two urn models which may also have independent interest.

Original language | English |
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Title of host publication | Extended Abstracts EuroComb 2021 |

Editors | Jaroslav Nešetřil |

Publisher | Springer Nature |

Pages | 129-135 |

Number of pages | 7 |

ISBN (Print) | 9783030838225 |

DOIs | |

State | Published - 2021 |

Event | European Conference on Combinatorics, Graph Theory and Applications 2021 - Barcelona, Spain Duration: 6 Sep 2021 → 10 Sep 2021 |

### Publication series

Name | Trends in Mathematics |
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Publisher | Springer |

Volume | 14 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Conference

Conference | European Conference on Combinatorics, Graph Theory and Applications 2021 |
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Country/Territory | Spain |

City | Barcelona |

Period | 6/09/21 → 10/09/21 |

### Bibliographical note

Funding Information:D. Hefetz—Research supported by ISF grant 822/18.

Publisher Copyright:

© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

## Keywords

- Games on graphs
- Random process