特殊正则图的度数博弈
Degree Game for Special Regular Graphs
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中文总结 AI 辅助
本文针对超立方体图、网格图等无限图族，改进了Maker-Breaker度数博弈中Breaker的度数下界，将原d/4的界限提升至至少⌊d/3⌋
中文摘要 AI 辅助
对于给定的d-正则图G，Maker-Breaker度数博弈由两名玩家交替认领G中此前未被认领的边。在标准变体中，Maker的目标是最大化其诱导子图的最大度数，而Breaker的目标是最小化该最大度数，等价于在自身子图中保证某一最小度数。经典配对策略表明，Breaker可在任意d-正则图的每个顶点处至少获得⌊d/4⌋条边。打破这一针对一般图乃至特定图类的界限，是组合博弈论中一个长期存在的开放问题；J. Beck在其专著中将这一挑战描述为位置博弈论七大最棘手开放问题之首。本文针对超立方体图Q_d、网格图、环面图等无限图族，改进了d/4的界限。我们首先证明，在Q_3的每个顶点处，Breaker可保证度数为1，随后将该结论推广至高维情形，即Breaker可保证度数至少为⌊d/3⌋。
英文摘要
For a given $d$-regular graph $G$, a Maker-Breaker degree game is played by two players who alternately claim previously unclaimed edges of $G$. In the standard variant, the goal of Maker is to maximize the maximum degree of their induced subgraph, while Breaker aims to minimize it, or equivalently, to guarantee a certain minimum degree in their own subgraph. A classic pairing strategy shows that Breaker can secure at least $\lfloor d/4 \rfloor$ edges at every vertex of any $d$-regular graph. Breaking this bound for general or even for specific classes of graphs has been a long-standing open problem in combinatorial game theory; indeed, J. Beck characterized this challenge in his monograph as the first among the seven most humiliating open problems of positional game theory. In this paper, we improve the $d/4$ bound for some infinite graph families, such as the hypercube graph $Q_d$, grids and tori. We first show that Breaker can secure a degree of one at every vertex in $Q_3$, then lift this to higher dimensions, where Breaker can guarantee a degree of at least $\lfloor d/3 \rfloor$.
