On edge-colouring-games by Erd\H{o}s, and Bensmail and Mc Inerney

We consider two games proposed by Erd\H{o}s, and one game by Bensmail and Mc Inerney, all with the same setup of two players alternately colouring one edge of a clique. We give observations and particular behaviour for each of these problems, and prove a first reduction towards confirming the conjecture by Bensmail and Mc Inerney. We state a conjecture for Erd\H{o}s' game on the largest induced maximum degree, and extensions to edge-transitive and, respectively, regular graphs.