Abstract
Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.
Highlights
Positional games belong to the family of perfect information games between two players, and they have become a field of intense studies throughout the last decades
Note that it makes sense to introduce a bias for Maker, but we will not consider this case in the remainder of our paper
We provide a different strategy for Waiter for each case
Summary
In the unbiased Waiter-Client pancyclicity game the following holds: τW C(PCn, 1) = n + (1 + o(1)) log n Note that this means that Waiter wins almost perfectly fast, as every spanning pancyclic subgraph of Kn has at least n + (1 − o(1)) log n edges [6]. Another game won by Maker in its unbiased version is the so-called connectivity game on Kn, introduced by Chvatal and Erdos [7] in which Maker’s goal is to claim any spanning subgraph of Kn. we already discussed that for large enough n, Maker has a strategy to create a Hamilton cycle asymptotically fast. Even when the maximum degree and Breaker’s bias are increasing with n, Maker has a strategy to win the fixed spanning tree game asymptotically fast. Any edge belonging to C ∪ W is said to be claimed, while all the other edges in play are called free
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