Semi-random Process Without Replacement

We introduce and study a semi-random multigraph process, which forms no-replacement variant of the process that was introduced in [3]. The starts with an empty graph on vertex set [n]. For every positive integers q \(1\le r\le n\), \(((q-1)n+r)\)th round decision-maker, called Builder, is offered \(\pi _q(r)\), where _1, \pi _2, \ldots \) sequence permutations \(S_n\), chosen independently uniformly at random. Builder then chooses additional (according to strategy his choice) connects it by edge _q(r)\).