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 $(p,q)=(1,3)$.

翻译：我们研究了Erdős提出的两种博弈以及Bensmail与Mc Inerney提出的一种博弈，这些博弈共享一个共同框架：两名玩家轮流对完全图的边进行着色，在偏向版本中，每轮分别着色$p$条和$q$条边，目标是通过各自诱导子图确定的图参数最大化。在无偏向情形中，我们首次给出归约方法以验证Bensmail与Mc Inerney的猜想，针对Erdős关于最大度的博弈提出新猜想，并将团与最大度版本推广至边传递图与正则图。在偏向情形中，最大度博弈与顶点捕获博弈已获解决，并证明了$(p,q)=(1,3)$时的团博弈结论。