Tree universality in positional games

In this paper we consider positional games where the winning sets are tree universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the complete graph \(K_n\), Maker has a strategy to occupy a graph which contains copies of all spanning trees with maximum degree at most \(c...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Adamski, Grzegorz, Antoniuk, Sylwia, Bednarska-Bzdęga, Małgorzata, Clemens, Dennis, Hamann, Fabian, Mogge, Yannick
Format: Article
Language:English
Subjects:
Games
Graph theory
Online Access:Get full text
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Summary:In this paper we consider positional games where the winning sets are tree universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the complete graph \(K_n\), Maker has a strategy to occupy a graph which contains copies of all spanning trees with maximum degree at most \(cn/\log(n)\), for a suitable constant \(c\) and \(n\) being large enough. We also prove an analogous result for Waiter-Client games. Both of our results show that the building player can play at least as good as suggested by the random graph intuition. Moreover, they improve on a special case of earlier results by Johannsen, Krivelevich, and Samotij as well as Han and Yang for Maker-Breaker games.
ISSN:2331-8422