Waiter-Client and Client-Waiter planarity, colorability and minor games
Dan Hefetz, Michael Krivelevich, Wei En Tan

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 , a family of sets and a positive integer , we consider two types of two player, perfect information games with no chance moves. In each round of the Waiter-Client game , the first player, called Waiter, offers the second player, called Client, elements of the board which have not been offered previously. Client then chooses one of these elements which he claims and the remaining elements to go back to Waiter. Waiter wins this game if by the time every element of has been claimed by some player, Client has claimed all elements of some ; 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 and Waiter wins otherwise. In this paper we…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
