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 a1(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 a1(G) for the disjoint union of graphs, prove a formula for a1(G) if G is triangle-free, and obtain the exact values of a1(Cn□Cm), a1(Cn□Pm), and a1(Pn□Pm) for all possible pairs (n,m).
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