正则图上的分支对策
Component Games on Regular Graphs
摘要：
研究了d-正则图边集上的（1：b）Maker-Breaker分量对策。Maker在这个游戏中的目标是建立一个大型的连接组件，而Breaker的目标是不让他这样做。对于Breaker的偏差b的所有值，我们确定Breaker是否获胜（在任何d-正则图上）或Maker是否获胜（在几乎每个d-正则图上），并为两个玩家提供明确的获胜策略。为此，我们证明了Gallai-Hasse-Roy-Vitaver关于没有长有向简单路的图方向的一个定理的推广。
英文摘要：
We study the (1:b) Maker-Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to not let him do so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players. To this end, we prove an extension of a theorem by Gallai-Hasse-Roy-Vitaver about graph orientations without long directed simple paths.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
