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

We study two games proposed by Erd\H{o}s, and one game by Bensmail and Mc Inerney, all sharing a common setup: two players alternately colour edges of a complete graph, or in the biased version, they colour $p$ and $q$ edges respectively on their turns, aiming to maximise a graph parameter determined by their respective induced subgraphs. In the unbiased case, we give a first reduction towards confirming the conjecture of Bensmail and Mc Inerney, propose a conjecture for Erd\H{o}s' game on maximum degree, and extend the clique and maximum-degree versions to edge-transitive and regular graphs. In the biased case, the maximum-degree and vertex-capturing games are resolved, and we prove the clique game with $(p,q)=(1,3)$.