Very fast construction of bounded-degree spanning graphs via the semi-random graph process

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 (Formula presented.) can be constructed with high probability in (Formula presented.) rounds. This is tight up to a multiplicative factor of (Formula presented.). 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.