在Waiter-Client游戏中创建生成树
Creating spanning trees in Waiter-Client games
AI总结：
研究完全图上公平Waiter-Client游戏中构建树副本的胜负策略，证明了基于树最大度的Waiter和Client的必胜条件，并扩展至Client-Waiter版本。
摘要：
对于正整数$n$和具有$n$个顶点的树$T_n$，我们考虑在完全图$K_n$上进行的一个公平的Waiter-Client游戏$textrm{WC}(n,T_n)$，其中Waiter的目标是迫使Client构建$T_n$的一个副本。我们证明，对于每个常数$c<1/3$，如果$Δ(T_n)le cn$且$n$足够大，那么Waiter在$textrm{WC}(n,T_n)$中有必胜策略。另一方面，我们证明存在一个正常数$c'<1/2$以及一族满足$Δ(T_n)le c'n$的树$T_n$，使得对于每个足够大的$n$，Client在$textrm{WC}(n,T_n)$游戏中都有必胜策略。我们还考虑了该游戏Client-Waiter版本中的对应问题。
英文摘要：
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 $Δ(T_n)\le cn$ 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 $Δ(T_n)\le c'n$ 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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