On edge-colouring-games by Erdős, and Bensmail and Mc Inerney

We consider two games proposed by Erdős, 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 Erdős' game on the largest induced maximum degree, and extensions to edge-transitive and, respectively, regular graphs.