Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multi graph process, which forms a no-replacement variant of the process that was introduced in [4]. The process starts with an empty graph on the vertex set [n]. For every positive integers q and 1 = r = n, in the ((q - 1)n + r)th round of the process, the decision-maker, called Builder, is offered the vertex p(q)(r), where p(1), p(2), ... is a sequence of permutations in Sn, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to p(q)(r).For several natural graph properties, such as k-connectivity, minimum degree at least k, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.