Positional games on random graphs

Details Positional games on random graphs Positional games on random graphs

2006-01-26

arxiv.org/abs/math/0601659v1

Random Structures & Algorithms 26 (2005), 204-223

Mathematics

Combinatorics

Scientific paper

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $F$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game.

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