  For a finite set $X$, a family of sets ${\mathcal F} \subseteq 2^X$ and a
positive integer $q$, we consider two types of two player, perfect information
games with no chance moves. In each round of the $(1 : q)$ Waiter-Client game
$(X, {\mathcal F})$, the first player, called Waiter, offers the second player,
called Client, $q+1$ elements of the board $X$ which have not been offered
previously. Client then chooses one of these elements which he claims and the
remaining $q$ elements to go back to Waiter. Waiter wins this game if by the
time every element of $X$ has been claimed by some player, Client has claimed
all elements of some $A \in {\mathcal F}$; otherwise Client is the winner.
Client-Waiter games are defined analogously, the main difference being that
Client wins the game if he manages to claim all elements of some $A \in
{\mathcal F}$ and Waiter wins otherwise. In this paper we study the
Waiter-Client and Client-Waiter versions of the non-planarity, $K_t$-minor and
non-$k$-colorability games. For each such game, we give a fairly precise
estimate of the unique integer $q$ at which the outcome of the game changes
from Client's win to Waiter's win. We also discuss the relation between our
results, random graphs, and the corresponding Maker-Breaker and
Avoider-Enforcer games.
