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arXiv · 1901.00351

$H$-games played on vertex sets of random graphs

Gal Kronenberg·Adva Mond·Alon Naor
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Abstract

We introduce a new type of positional games, played on a vertex set of a graph. Given a graph $G$, two players claim vertices of $G$, where the outcome of the game is determined by the subgraphs of $G$ induced by the vertices claimed by each player (or by one of them). We study classical positional games such as Maker-Breaker, Avoider-Enforcer, Waiter-Client and Client-Waiter games, where the board of the game is the vertex set of the binomial random graph $G\sim G(n,p)$. Under these settings, we consider those games where the target sets are the vertex sets of all graphs containing a copy of a fixed graph $H$, called $H$-games, and focus on those cases where $H$ is a clique or a cycle. We show that, similarly to the edge version of $H$-games, there is a strong connection between the threshold probability for these games and the one for the corresponding vertex Ramsey property (that is, the property that every $r$-vertex-coloring of $G(n,p)$ spans a monochromatic copy of $H$). Another similarity to the edge version of these games we demonstrate, is that the games in which $H$ is a triangle or a forest present a different behavior compared to the general case.

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Gal Kronenberg, Adva Mond, Alon Naor. 2019-01-02. $H$-games played on vertex sets of random graphs. https://arxiv.org/abs/1901.00351

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