Abstract

AbstractIn a Maker‐Breaker game on a graph G, Breaker and Maker alternately claim edges of G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker‐Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between n points chosen uniformly at random in the unit square by order of increasing edge‐length then, with probability tending to one as n →∞, the graph becomes Maker‐win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the H‐game as soon as there is a subgraph from a finite list of “minimal graphs.” These results also allow us to give precise expressions for the limiting probability that G(n, r) is Maker‐win in each case, where G(n, r) is the graph on n points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most r. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 553–607, 2014

Highlights

  • We explicitly determined the hitting radius for the games of connectivity, perfect matching and Hamilton cycle, all played on the edges of the random geometric graph

  • For the connectivity game it is the minimum degree two, in the case of the perfect matching game it is again the minimum degree two accompanied by the minimum edge degree three, and for the Hamilton cycle game we have the minimum degree four

  • In the connectivity game the hitting time for Maker-win is the same as for the minimum degree two [23], the same holds for the perfect matching game [4], and the condition changes to minimum degree four for the Hamilton cycle game [4]

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Summary

Introduction

Several works followed, including [11, 4], resulting in precise descriptions of the limiting probabilities for Maker-win in games of connectivity, perfect matching and Hamilton cycle, all played on the edges of a random graph G ∼ G(n, p). Theorem 1.1 The random geometric graph process satisfies ρn(Maker wins the connectivity game) = ρn(minimum degree ≥ 2) whp. The theorem on the hitting radius allows us to determine the hitting probability This time we make use of a theorem of Penrose [22] (stated as Theorem 2.9 below) on the appearance of small subgraphs to obtain: Corollary 1.8 Let H be any fixed graph, and let kH denote the smallest k for which the H-game is Maker-win on a k-clique. P(Maker wins the H-game) → f (c), where 0 < f (c) < 1 (is an expression which can be computed explicitly in principle and) satisfies f (c) → 0 as c → −∞; and f (c) → 1 as c → ∞

Overview
Preliminaries
Probabilistic preliminaries
Geometric preliminaries
The Dissection Lemma
The Obstruction Lemma
The connectivity game
The Hamilton cycle game
Two helpful path games
Blob Cycles
Proof of the Hamilton cycle game
The perfect matching game
The H-game
Findings
Conclusion and further work
Full Text
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