Tree universality in positional games

Journal
Combinatorics, probability & computing  
Volume
34
Issue
3
Start Page
338
End Page
358
Citation
Combinatorics Probability and Computing 34 (3): 338-358 (2025)
Publisher DOI
10.1017/S0963548324000397
Scopus ID
2-s2.0-85212559258
Publisher
Cambridge Univ. Press
ISSN
09635483
In this paper we consider positional games where the winning sets are edge sets of tree-universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the edges of the complete graph <![CDATA[ $K_n$ ]]>, Maker has a strategy to claim a graph which contains copies of all spanning trees with maximum degree at most <![CDATA[ $cn/\log (n)$ ]]>, for a suitable constant <![CDATA[ $c$ ]]> and <![CDATA[ $n$ ]]> 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.
Subjects
embedding | expander | Maker-Breaker games | tree | universality | Waiter-Client games
DDC Class
600: Technology