1. Creating spanning trees in Waiter-Client games
- Author
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Adamski, Grzegorz, Antoniuk, Sylwia, Bednarska-Bzdęga, Małgorzata, Clemens, Dennis, Hamann, Fabian, and Mogge, Yannick
- Subjects
Mathematics - Combinatorics ,05C05, 91A24 - Abstract
For a positive integer $n$ and a tree $T_n$ on $n$ vertices, we consider an unbiased Waiter-Client game $\textrm{WC}(n,T_n)$ played on the complete graph~$K_n$, in which Waiter's goal is to force Client to build a copy of $T_n$. We prove that for every constant $c<1/3$, if $\Delta(T_n)\le cn$ and $n$ is sufficiently large, then Waiter has a winning strategy in $\textrm{WC}(n,T_n)$. On the other hand, we show that there exist a positive constant $c'<1/2$ and a family of trees $T_{n}$ with $\Delta(T_n)\le c'n$ such that Client has a winning strategy in the $\textrm{WC}(n,T_n)$ game for every $n$ sufficiently large. We also consider the corresponding problem in the Client-Waiter version of the game.
- Published
- 2024