Maker-Breaker游戏的拉姆齐型结果

中文摘要 AI 辅助

若图G的每条边的2-染色都包含H的单色拷贝，且G的每个真子图都存在不包含此类单色拷贝的2-染色，则称G是图H的极小拉姆齐图。刻画极小拉姆齐图是广受研究的问题，该领域近期研究包括刻画所有H的极小拉姆齐图构成的集合ℳ₂(H)的大小，或求ℳ₂(H)中所有图的最小最小度。本文通过考虑图G上的Maker-Breaker H-游戏，引入上述概念的博弈论类比：游戏中Maker和Breaker轮流认领G的未认领边，若最终Maker的边构成的图包含H的拷贝则Maker获胜，否则Breaker获胜。若Maker在G上的Maker-Breaker H-游戏中有获胜策略，则称G对H是可胜的；若G的每个真子图上Breaker都能在H-游戏中获胜，则称G是极小可胜的。沿极小拉姆齐理论的思路，本文刻画了存在无穷多极小可胜图的所有图H，并证明了这些极小可胜图中所有图的最小最小度的紧界，还针对树、环、团、完全二部图获得了精确结果。总体而言，本文发现拉姆齐设定与Maker-Breaker设定存在诸多相似性，但也存在显著差异。

英文摘要

A graph $G$ is minimal Ramsey for a graph $H$ if every $2$-colouring of the edges of $G$ contains a monochromatic copy of $H$, but for every proper subgraph of $G$, there is a $2$-colouring that does not contain such a monochromatic copy. Characterizing minimal Ramsey graphs is a widely studied problem. Recent research in this field includes the characterization of the size of the set $\mathcal{M}_2(H)$ of all minimal Ramsey graphs for $H$, or finding the smallest minimum degree among all graphs in $\mathcal{M}_2(H)$. In this paper, we introduce a game theoretic analogue of the above concept by considering the Maker-Breaker $H$-game on a graph $G$. In this game, two players, Maker and Breaker, alternately claim unclaimed edges of $G$, and Maker wins if in the end of the game the graph spanned by Maker's edges contains a copy of $H$. Otherwise, Breaker wins the game. We call a graph $G$ winnable for $H$ if Maker has a winning strategy for the Maker-Breaker $H$-game on $G$, and we call it minimal winnable if additionally Breaker wins the $H$-game on every proper subgraph of $G$. Along the lines of minimal-Ramsey theory, we characterize all graphs $H$ for which there exist infinitely many minimal winnable graphs, and we prove tight bounds for the smallest minimum degree among all these minimal winnable graphs. Amongst others, we obtain precise results for trees, cycles, cliques, and complete bipartite graphs. In general, we find many similarities between the Ramsey setting and the Maker-Breaker setting, but we also show substantial differences.

发表机构

• Hamburg University of Technology（汉堡工业大学）

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