Client-Waiter Games on Complete and Random Graphs

For a graph $ G $, a monotone increasing graph property $ \mathcal{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...

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Veröffentlicht in:The Electronic journal of combinatorics 2016-12, Vol.23 (4)
Hauptverfasser: Dean, Oren, Krivelevich, Michael
Format: Artikel
Sprache:eng
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Zusammenfassung:For a graph $ G $, a monotone increasing graph property $ \mathcal{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 $ \mathcal{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.
ISSN:1077-8926
1077-8926
DOI:10.37236/6039