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 multigraph process, which forms a no-replacement variant of the process that was introduced in \cite{BHKPSS}. The process starts with an empty graph on the vertex set $[n]$. For every positive integer $k$, in the $k$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $v_k := \pi_{\lceil k/n \rceil}(k - \lfloor (k-1)/n \rfloor n)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, 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 $v_k$. 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 two urn models which may also have independent interest.