The Waiter-Client maximum degree with bias $q$ is played by Waiter and Client on the edges of the complete graph $K_n$ on $n$ vertices. In each round of the game, Waiter offers Client $q+1$ previously unoffered edges, Client claims one of them, and the rest go to Waiter. Client's goal is to minimize the maximum degree in the graph of his edges by the end of the game.
We determine the asymptotic value of Client's maximum degree for all values of the bias $q(n)$ outside the critical region $q(n)=\Theta(n/\log n)$. In the talk we will mostly focus on the very important unbiased case $q=1$, for which we show that, assuming the perfect play of both players, Client's maximum degree $D$ satisfies: $D=n/2+\Theta(\sqrt{n\log n})$. The obtained results comply very well with the probabilistic intuition, derived from observing the typical maximum degree of the random graph $G(n,m)$ with the corresponding value of of $m(n)$: $m=n^2/2(q+1)$.
Joint work with M. Krivelevich.