Thresholds for the monochromatic clique transversal game

We study a recently introduced two-person combinatorial game, the (a,b)-monochromatic clique transversal game which is played by Alice and Bob on graph G. As we observe, this equivalent to (b,a)-biased Maker–Breaker clique-hypergraph of Our main results concern threshold bias a1(G) that smallest integer such can win in (a,1)-monochromatic G if she first play. Among other results, determine possible values for disjoint union graphs, prove formula triangle-free, obtain exact a1(Cn□Cm), a1(Cn□Pm), a1(Pn□Pm) all pairs (n,m).