TY - CHAP

T1 - Semi-random Process Without Replacement

AU - Gilboa, Shoni

AU - Hefetz, Dan

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Games on graphs

KW - Random process

UR - http://www.scopus.com/inward/record.url?scp=85114101766&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_21

DO - 10.1007/978-3-030-83823-2_21

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AN - SCOPUS:85114101766

T3 - Trends in Mathematics

SP - 129

EP - 135

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -