论文标题
单色集团横向游戏的阈值
Thresholds for the monochromatic clique transversal game
论文作者
论文摘要
我们研究了最近推出的两人组合游戏，即$（a，b）$ - 单色集团横向游戏，由爱丽丝和鲍勃在图形$ g $上玩。正如我们观察到的那样，此游戏等同于$（b，a）$ - 偏见的制造商突破性游戏，在$ g $的集合hypergraph上玩。我们的主要结果涉及最小的整数$ a $ a_1（g）$的阈值偏差$ a_1（g）$，因此，如果她是第一个玩的话，爱丽丝可以在$ g $上赢得$（a，1）$ - 单色clique transerssal游戏。除其他结果外，我们确定图表不相交的$ a_1（g）$的可能值，如果$ g $不含三角形，则证明了$ a_1（g）$的公式，并获得$ a_1的确切值（c_n \，\ square \ square \ square \，c_m）$，$ a_1（$ a_1（c_n \ a _1） \，\ square \，p_m）$对于所有可能的对$（n，m）$。
We study a recently introduced two-person combinatorial game, the $(a,b)$-monochromatic clique transversal game which is played by Alice and Bob on a graph $G$. As we observe, this game is equivalent to the $(b,a)$-biased Maker-Breaker game played on the clique-hypergraph of $G$. Our main results concern the threshold bias $a_1(G)$ that is the smallest integer $a$ such that Alice can win in the $(a,1)$-monochromatic clique transversal game on $G$ if she is the first to play. Among other results, we determine the possible values of $a_1(G)$ for the disjoint union of graphs, prove a formula for $a_1(G)$ if $G$ is triangle-free, and obtain the exact values of $a_1(C_n \,\square\, C_m)$, $a_1(C_n \,\square\, P_m)$, and $a_1(P_n \,\square\, P_m)$ for all possible pairs $(n,m)$.
