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图的对策匹配数

Game matching number of graphs

Daniel W. Cranston, William B. Kinnersley, Suil O., Douglas B. West

备注：
16 pages; this version improves explanations at a number of points and adds a few more examples
发表信息：
Discrete Applied Math. Vol. 161(13-14), 2013, pp. 1828-1836
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摘要：

我们研究的竞争优化版本的$α'(G)$，在图$G$匹配的最大尺寸。玩家交替添加$G$的边到匹配，直到它成为最大匹配。一个参与者（Max）希望匹配大，另一个（Min）希望匹配小。当Max或Min开始时，在最佳播放下得到的大小分别表示为$\Max(G)$和$\Min(G)$。我们表明，总是$|\Max(G)-\Min(G)|\le 1$。我们得到了$\Max(G)=α'(G)$在Carnival乘积下保持不变的一个充分条件。一般来说，$\Max(G)\ge \frac23α'(G)$，与平等的许多分裂图，而$\Max(G)\ge\frac34α'(G)$当$G$是一个森林。当$G$是一个3-正则$n$ -顶点连通图时，$\Max(G) \ge n/3$，并且有这样的例子$\Max(G)\le 7n/18$ .对于$n$ -顶点路径或循环，答案大致为$n/7$。

英文摘要：

We study a competitive optimization version of $α'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted $\Max(G)$ and $\Min(G)$, respectively. We show that always $|\Max(G)-\Min(G)|\le 1$. We obtain a sufficient condition for $\Max(G)=α'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23α'(G)$, with equality for many split graphs, while $\Max(G)\ge\frac34α'(G)$ when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, $\Max(G) \ge n/3$, and there are such examples with $\Max(G)\le 7n/18$. For an $n$-vertex path or cycle, the answer is roughly $n/7$.

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。

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