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arXiv · 2403.18534

Creating spanning trees in Waiter-Client games

Grzegorz Adamski·Sylwia Antoniuk·Małgorzata Bednarska-Bzdęga·Dennis Clemens·Fabian Hamann·Yannick Mogge
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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.

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Grzegorz Adamski, Sylwia Antoniuk, Małgorzata Bednarska-Bzdęga, Dennis Clemens, Fabian Hamann, Yannick Mogge. 2024-03-27. Creating spanning trees in Waiter-Client games. https://arxiv.org/abs/2403.18534

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