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Multistage positional games

Publikationstyp
Journal Article
Date Issued
2023-09
Sprache
English
Author(s)
Barkey, Juri 
Clemens, Dennis  orcid-logo
Hamann, Fabian  orcid-logo
Mikalački, Mirjana  
Sgueglia, Amedeo  
Institut
Mathematik E-10  
TORE-URI
http://hdl.handle.net/11420/15378
Journal
Discrete mathematics  
Volume
346
Issue
9
Article Number
113478
Citation
Discrete Mathematics 346 (9): 113478 (2023-09)
Publisher DOI
10.1016/j.disc.2023.113478
Scopus ID
2-s2.0-85153621421
ArXiv ID
2202.04344v2
Publisher
Elsevier
We initiate the study of a new variant of the Maker-Breaker positional game, which we call multistage game. Given a hypergraph H=(X,F) and a bias b≥1, the (1:b) multistage Maker-Breaker game on H is played in several stages as follows. Each stage is played as a usual (1:b) Maker-Breaker game, until all the elements of the board get claimed by one of the players, with the first stage being played on H. In every subsequent stage, the game is played on the board reduced to the elements that Maker claimed in the previous stage, and with the winning sets reduced to those fully contained in the new board. The game proceeds until no winning sets remain, and the goal of Maker is to prolong the duration of the game for as many stages as possible. In this paper we estimate the maximum duration of the (1:b) multistage Maker-Breaker game, for biases b subpolynomial in n, for some standard graph games played on the edge set of Kn: the connectivity game, the Hamilton cycle game, the non-k-colorability game, the pancyclicity game and the H-game. While the first three games exhibit a probabilistic intuition, it turns out that the last two games fail to do so.
Subjects
Beck's criterion
Games on graphs
Maker-Breaker games
Multistage games
DDC Class
510: Mathematik
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