AI 中文总结
本文研究正则图上的选择者-挑选者度数博弈，突破经典策略的d/4界限，证明挑选者可确保3-正则图顶点至少1度、任意d-正则图顶点至少⌊d/3⌋度，揭示挑选者在稀疏局部博弈中的结构优势。
AI 中文摘要
在正则图边集上进行的无偏选择者-挑选者（又称客户端-等待者）博弈中，挑选者每回合提供一对未被占有的边，选择者认领其中一条，剩余边归挑选者所有。我们研究在d-正则图上进行的选择者-挑选者（C-P）度数博弈，其中选择者的目标是最大化其导出子图的最大度数，而挑选者的目标是通过在自身子图中确保某个最小度数来防御每个顶点。尽管经典的静态配对策略在一般d-正则图上的 Maker-Breaker（M-B）博弈中为破坏者、在C-P博弈中为挑选者保证了至少⌊d/4⌋的最小度数，在这两种博弈框架中突破该阈值一直是主要的开放挑战。根据J. Beck的基础专著，该挑战是组合博弈论中七个最棘手问题中的第一个。我们的主要结果是挑选者可以突破d/4的界限。首先，我们证明在任何3-正则图上，挑选者总能确保每个顶点至少有1度。基于此，我们引入一种直接策略，证明对于任何d-正则图，挑选者可以确保每个顶点至少有⌊d/3⌋的度数。这凸显了在稀疏局部博弈中，挑选者通常比破坏者具有的基本结构优势。
英文摘要
In the unbiased Chooser-Picker (also known as Client-Waiter) game played on the edge set of a graph, Picker offers a pair of unclaimed edges in each turn, Chooser claims one, and the remaining edge goes back to Picker. We study the Chooser-Picker (C-P) degree game played on $d$-regular graphs, where Chooser aims to maximize the maximum degree of their induced subgraph, and Picker's objective is to defend every vertex by securing a certain minimum degree in Picker's own subgraph. While classical static pairing strategies guarantee a minimum degree of at least $\lfloor d/4 \rfloor$ for Breaker on general $d$-regular graphs in Maker-Breaker (M-B) games and for Picker in C-P games, outperforming this threshold has been a major open challenge in both frameworks. According to the foundational monograph of J. Beck, this challenge stands as the first among the seven most humiliating problems in combinatorial game theory. Our main result is that Picker can beat the $d/4$ bound. First, we prove that Picker can always guarantee a degree of at least one at every vertex on any $3$-regular graph. Based upon this we introduce a direct strategy to prove that Picker can secure a degree of at least $\lfloor d/3 \rfloor$ at every vertex for any $d$-regular graph. This highlights a fundamental structural advantage that Picker usually possesses over Breaker in sparse local games.
