Waiter-Client and Client-Waiter planarity, colorability and minor games

TL;DR

This paper analyzes positional games played on finite sets related to graph properties like planarity, minors, and colorability, providing thresholds for game outcomes and connecting these results to random graphs and other combinatorial games.

Contribution

It introduces and studies Waiter-Client and Client-Waiter versions of graph property games, offering precise thresholds for game outcome transitions and relating them to existing graph and positional game theories.

Findings

Determined thresholds for game outcome changes from Client to Waiter.

Established connections between these games and random graph models.

Compared results with Maker-Breaker and Avoider-Enforcer games.

Abstract

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.