In the (1:b) component game played on a graph G, two players, Maker and Breaker, alternately claim 1 and b previously unclaimed edges of G, respectively. Maker’s aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty (b +2)-core in the graph.

For any integer k, the k-core of a graph is its largest subgraph of minimum degree at least k. Pittel, Spencer and Wormald showed in 1996 that for any k ≥ 3 there exists a constant c_k, which they determine, such that p = c_k/n is the threshold function for the appearance of the k-core in \(G \sim {\cal G}\left( {n,p} \right)\). More precisely, \(G \sim {\cal G}\left( {n,c/n} \right)\) has WHP a linear-size k-core for constant c>c_k, and an empty k-core when c<c_k.

We show that for any positive constant integer b, when playing the (1:b) component game on \(G \sim {\cal G}\left( {n,c/n} \right)\), Maker can WHP build a linear-size component if c > c_b+2, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if c< c_b+2.

For the strategy of Maker when c> c_b+2, we utilise known results on the k-core. For Breaker when c< c_b+2, we make use of a result of Achlioptas and Molloy (sketching its proof) that states that after deleting all vertices of degree less than k, and repeating this step a constant number of times, G is WHP shattered into pieces of polylogarithmic size.

在图G上进行的 (1: b ) 组件博弈中，两个玩家，M aker和 B reaker，分别交替声明G的 1 和b先前未声明的边。M aker的目标是最大化她图中最大连接组件的大小，而 B reaker试图最小化它。我们表明，二项式随机图上的博弈结果与图中非空 ( b +2) 核的出现密切相关。

对于任何整数k，图的k核是其最小度数至少为k的最大子图。Pittel 、Spencer 和 Wormald在1996 年表明，对于任何k ≥ 3，存在一个常数c _k，他们确定了这个常数，因此p = c _k / n是\(G \ sim {\cal G}\left( {n,p} \right)\)。更准确地说，\(G \sim {\cal G}\left( {n,c/n} \right)\)具有 WHP 一个线性大小的k核，用于常数c > c _k，以及当c < c _k时为空的k核。

我们证明对于任何正常数整数b ，当在\(G \sim {\cal G}\left( {n,c/n} \right)\)上玩 (1: b ) 分量游戏时，Maker可以如果c > c b _{+ 2} ， WHP 构建一个线性大小的组件，而如果c < c b _{+ 2}，B reaker可以阻止 M aker构建大于多对数大小的组件。

对于当c > c _{b +2时 M} aker的策略，我们利用k核上的已知结果。对于 B reaker，当c < c _{b +2}时，我们利用 Achlioptas 和 Molloy 的结果（草拟其证明），即在删除所有度数小于k的顶点后，并重复此步骤恒定次数，G是 WHP 破碎成多对数大小的碎片。