We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced in . 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 πq(r), where π1, π2, … is a sequence of permutations in Sn, 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 πq(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.
|Title of host publication||Extended Abstracts EuroComb 2021|
|Number of pages||7|
|State||Published - 2021|
|Event||European Conference on Combinatorics, Graph Theory and Applications 2021 - Barcelona, Spain|
Duration: 6 Sep 2021 → 10 Sep 2021
|Name||Trends in Mathematics|
|Conference||European Conference on Combinatorics, Graph Theory and Applications 2021|
|Period||6/09/21 → 10/09/21|
Bibliographical noteFunding Information:
D. Hefetz—Research supported by ISF grant 822/18.
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
- Games on graphs
- Random process