Semi-random Process Without Replacement

Shoni Gilboa, Dan Hefetz

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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 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.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages7
StatePublished - 2021

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • Games on graphs
  • Random process


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