In this paper, we study the following recently proposed semi‐random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded‐degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n‐vertex graph with maximum degree can be constructed with high probability in rounds. This is tight up to a multiplicative factor of . We also obtain tight bounds for the number of rounds necessary to embed bounded‐degree spanning trees, and consider a nonadaptive variant of this setting.

中文翻译：

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通过半随机图过程非常快速地构建有界度生成图

在本文中，我们研究了以下最近提出的半随机图过程：从n个顶点上的一个空图开始，该过程以回合的方式进行，其中在每个回合中，我们获得一个统一的随机顶点v，并且必须立即（在在线方式）将带v的边沿事件添加到我们的图形中。最终目标是使构造的图满足某些预定的单调图特性。Alon询问是否可以在O（n）轮中以高概率构造每个给定的有界度跨度图。我们从很强的意义上肯定地回答了这个问题，表明可以最大可能地构造任何最大度数的n顶点图。回合。这与的乘法因子紧密相关。我们还获得了嵌入有界度的生成树所需的轮数的严格边界，并考虑了此设置的非自适应变体。