Semi-random Process Without Replacement

Abstract

We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced in [3]. The process starts with an empty graph on the vertex set [n]. For every positive integers q and \(1\le r\le n\), in the \(((q-1)n+r)\)th round of the process, the decision-maker, called Builder, is offered the vertex \(\pi _q(r)\), 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 \(\pi _q(r)\).