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 $a_1\left(G\right)$ 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 $a_1\left(G\right)$ for the disjoint union of graphs, prove a formula for $a_1\left(G\right)$ 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 $(n,m)$.

我们研究最近推出的两人组合游戏，$(A,b)$- Alice 和 Bob 在图上玩的单色集团横向游戏$G$. 正如我们观察到的，这个游戏相当于$(b,A)$-在 clique-hypergraph 上玩的有偏见的 Maker-Breaker 游戏$G$. 我们的主要结果涉及阈值偏差$A_{\mathrm{1个}}\left(G\right)$那是最小的整数$A$这样爱丽丝就可以赢得$(A,\mathrm{1个})$- 单色集团横向游戏$G$如果她是第一个玩的。在其他结果中，我们确定的可能值$A_{\mathrm{1个}}\left(G\right)$对于图的不相交并，证明一个公式$A_{\mathrm{1个}}\left(G\right)$如果$G$是无三角形的，并获得的精确值$A_{\mathrm{1个}}(C_n\square C_{米})$,$A_{\mathrm{1个}}(C_n\square P_{米})$， 和$A_{\mathrm{1个}}(P_n\square P_{米})$对于所有可能的对$(n,米)$.