Biased Games On Random Boards
Asaf Ferber, Roman Glebov, Michael Krivelevich, Alon Naor

TL;DR
This paper analyzes biased Maker-Breaker and Avoider-Enforcer games on random graphs, establishing the asymptotic critical bias for various positional games and settling a related conjecture.
Contribution
It determines the asymptotic critical bias for multiple biased positional games on random graphs, confirming a conjecture and extending understanding of game thresholds.
Findings
Critical bias for Maker-Breaker games is asymptotically np/ln n for p=ω(ln n/n)
Critical bias for Maker-Breaker games is Θ(np/ln n) when p=Θ(ln n/n)
Critical bias for Avoider-Enforcer games is Θ(np/ln n) for p=Ω(ln n/n)
Abstract
In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board . In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of and Breaker responds by claiming previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the -vertex-connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with a slight modification, where the two players claim at least 1 and at least previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker-Breaker games are known to be "bias-monotone", that is, if Maker wins the…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Stochastic processes and statistical mechanics
