Hitting time results for Maker‐Breaker games

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document} **image** . We focus on three natural target properties for Maker's graph, namely being k -vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k -vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojakovic and Szabo. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011, © 2012 Wiley Periodicals, Inc. (Supported by USA-Israel BSF (2006322); Israel Science Foundation (1063/08); Farajun Foundation Fellowship.)

[1]  Michael Krivelevich,et al.  Global Maker-Breaker games on sparse graphs , 2011, Eur. J. Comb..

[2]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[3]  Michael Krivelevich,et al.  On two Hamilton cycle problems in random graphs , 2008 .

[4]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[5]  A. Frieze.  ON MATCHINGS AND HAMILTON CYCLES IN RANDOM GRAPHS , 1988 .

[6]  János Komlós,et al.  Limit distribution for the existence of Hamiltonian cycles in a random graph , 2006, Discret. Math..

[7]  Michael Krivelevich,et al.  Winning Fast in Sparse Graph Construction Games , 2008, Comb. Probab. Comput..

[8]  Tibor Szabó,et al.  Positional games on random graphs , 2005 .

[9]  Michael Krivelevich,et al.  Hitting time results for Maker-Breaker games: extended abstract , 2011, SODA '11.

[10]  Milos Stojakovic.  Games on Graphs , 2014, ICCS.

[11]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[12]  P. Erdös,et al.  Biased Positional Games , 1978 .

[13]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[14]  E. Chong,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[15]  F. C. Santos,et al.  Games on Graphs , 2006 .

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  J. Pach,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[18]  J. Beck.  Combinatorial Games: Tic-Tac-Toe Theory , 2008 .

[19]  Benny Sudakov,et al.  On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs , 2011, SIAM J. Discret. Math..

[20]  Aleksandar Pekec,et al.  A Winning Strategy for the Ramsey Graph Game , 1995, Combinatorics, Probability and Computing.

[21]  S. Janson,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[22]  Tibor Szabó,et al.  Positional games on random graphs , 2006, Random Struct. Algorithms.

[23]  Alfred Lehman,et al.  A Solution of the Shannon Switching Game , 1964 .

[24]  Paul Erdös,et al.  On a Combinatorial Game , 1973, J. Comb. Theory A.

[25]  Michael Krivelevich,et al.  Fast winning strategies in Maker-Breaker games , 2009, J. Comb. Theory, Ser. B.

[26]  Sebastian U. Stich,et al.  On Two Problems Regarding the Hamiltonian Cycle Game , 2009, Electron. J. Comb..

[27]  E. M. Palmer,et al.  Hitting time fork edge-disjoint spanning trees in a random graph , 1995 .

[28]  Michael Krivelevich,et al.  A sharp threshold for the Hamilton cycle Maker–Breaker game , 2009 .

[29]  Michael Krivelevich,et al.  A sharp threshold for the Hamilton cycle Maker–Breaker game , 2009, Random Struct. Algorithms.

[30]  Benny Sudakov,et al.  Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs , 2009, Combinatorics, Probability and Computing.

[31]  B. Bollobás,et al.  Random Graphs of Small Order , 1985 .

[32]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[33]  Michael Krivelevich,et al.  Positional Games , 2014, 1404.2731.