Client-Waiter games on complete and random graphs

TL;DR

This paper investigates Client-Waiter positional games on complete and random graphs, providing bounds on critical biases and comparing outcomes with Waiter-Client and Maker-Breaker games, including extensions to biased games and trees.

Contribution

It introduces and analyzes Client-Waiter games on various graph structures, establishing bounds and extending known results from Maker-Breaker games.

Findings

Bounds on critical bias for large star, long path, and large component games.

Client-Waiter results align with Maker-Breaker outcomes on random graphs.

Extensions of results to biased games and tree structures.

Abstract

For a graph G, a monotone increasing graph property P and positive integer q, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most q+1 unclaimed edges of G from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property P by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the H-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the H-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.