Waiter–Client and Client–Waiter games

Tan, Wei En (2017). Waiter–Client and Client–Waiter games. University of Birmingham. Ph.D.

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Abstract

In this thesis, we consider two types of positional games; \(Waiter\)-\(Client\) and \(Client\)-\(Waiter\) games. Each round in a biased (\(a\):\(b\)) game begins with Waiter offering a+b free elements of the board to Client. Client claims \(a\) elements among these and the remaining \(b\) elements are claimed by Waiter. Waiter wins in a Waiter-Client game if he can force Client to fully claim a \(winning\) \(set\), otherwise Client wins. In a Client-Waiter game, Client wins if he can claim a winning set himself, else Waiter wins.

We estimate the \(threshold\) \(bias\) of four different (\(1\):\(q\)) Waiter-Client and Client-Waiter games. This is the unique value of Waiter's bias \(q\) at which the player with a winning strategy changes. We find its asymptotic value for both versions of the complete-minor and non-planarity games and give bounds for both versions of the non-\(r\)-colourability and \(k\)-SAT games. Our results show that these games exhibit a heuristic called the \(probabilistic\) \(intuition\).

We also find sharp probability thresholds for the appearance of a graph in the random graph \(G\)(\(n\),\(p\)) on which Waiter and Client win the (\(1\):\(q\)) Waiter-Client and Client-Waiter Hamiltonicity games respectively.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):