平衡游戏之探索

平衡游戏在图 G 上进行，由两位玩家 Admirable (A) 和 Impish (I) 轮流选择无标签的顶点。Admirable 将选定的顶点标记为 0，Impish 标记为 1， resulting label on any edge is the sum modulo 2 of the labels of the vertices incident to that edge。 Let e_0 and e_1 denote the number of edges labeled by 0 and 1 after all the vertices are labeled。平衡游戏的偏差定义为 d = e_1 - e_0。两个玩家有着反目标: Admirable attempts to minimize the discrepancy d while Impish attempts to maximize d。当 (A) first move in the game 时， (A)-start game balance number, b^A_g(G)，是 d 的值 when both players play optimally；当 (I) makes the first move in the game 时， (I)-start game balance number, b^I_g(G)，是 d 的值 when both players play optimally。Among other results, we show that if G has order n, then -log_2(n) ≤ b^A_g(G) ≤ n/2 if n is even and 0 ≤ b^A_g(G) ≤ n/2 + log_2(n) if n is odd。Moreover we show that b^A_g(G) + b^I_g(\overline{G}) = floor(n/2)。