Thresholds for the monochromatic clique transversal game

IF 0.8 4区数学 Q2 MATHEMATICS Expositiones Mathematicae Pub Date : 2023-03-01 DOI:10.1016/j.exmath.2022.11.001

Csilla Bujtás , Pakanun Dokyeesun , Sandi Klavžar

引用次数: 1

Abstract

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 $a_{1} (C_{n} □ C_{m})$ , $a_{1} (C_{n} □ P_{m})$ , and $a_{1} (P_{n} □ P_{m})$ for all possible pairs $(n, m)$ .

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单色集团横向博弈的阈值

最近介绍二人组合游戏,我们研究(a, b)单色集团横向游戏由爱丽丝和鲍勃在一个图G .我们观察,这个游戏相当于(b, a)偏见Maker-Breaker游戏的clique-hypergraph G .我们的主要结果担心阈值偏差a1 (G)是最小的整数,爱丽丝可以赢得(a, 1)单色集团横向游戏G如果她是第一次玩。在其他结果中，我们确定了图的不相交并的a1(G)的可能值，证明了如果G是无三角形的a1(G)的一个公式，并获得了所有可能对(n,m)的a1(Cn□Cm)， a1(Cn□Pm)和a1(Pn□Pm)的精确值。

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来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

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