The -process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The -process generalizes both the Achlioptas process and the semi-random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the -process. Using this condition, in the semi-random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben-Eliezer et al. (RSA, 2020).

中文翻译：

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自适应随机图过程中的尖锐阈值

这-process 是一个单人游戏，其中玩家最初会看到空图顶点。在每个步骤中，边的子集根据分布独立采样。然后玩家选择一条边从，并添加到其当前图表。对于固定单调递增图属性，玩家的目标是迫使图表满足尽可能少的步骤。这-process 概括了 Achlioptas 过程和半随机图过程。我们证明了尖锐阈值存在的充分条件在里面-过程。利用这个条件，在半随机过程中，我们证明了当对应于哈密顿量或包含完美匹配。这解决了 Ben-Eliezer 等人提出的两个悬而未决的问题。 （RSA，2020）。