The random graph intuition for the tournament game

Clemens, Dennis; Gebauer, Heidi; Liebenau, Anita

doi:10.1017/S096354831500019X

Please use this identifier to cite or link to this item: https://doi.org/10.21256/zhaw-15843

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	The random graph intuition for the tournament game
Authors:	Clemens, Dennis Gebauer, Heidi Liebenau, Anita
DOI:	10.1017/S096354831500019X 10.21256/zhaw-15843
Published in:	Combinatorics, Probability and Computing
Volume(Issue):	25
Issue:	1
Page(s):	76
Pages to:	88
Issue Date:	2015
Publisher / Ed. Institution:	Cambridge University Press
ISSN:	0963-5483 1469-2163
Language:	English
Subjects:	Mathematics - combinatorics
Subject (DDC):	510: Mathematics
Abstract:	In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph on n vertices and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2-o(1))log_2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two "clever" players and the game played by two "random" players, is supported by the tournament game: It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation-game version of the tournament game, where Maker wins the game if the final digraph – containing also the edges directed by Breaker – possesses a copy of T_k. We prove that in that game Breaker has a winning strategy for k = (4+o(1))log_2 n.
Further description:	Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)
URI:	https://digitalcollection.zhaw.ch/handle/11475/15843
Fulltext version:	Published version
License (according to publishing contract):	Licence according to publishing contract
Departement:	School of Engineering
Organisational Unit:	Institute of Applied Mathematics and Physics (IAMP)
Appears in collections:	Publikationen School of Engineering

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Clemens, D., Gebauer, H., & Liebenau, A. (2015). The random graph intuition for the tournament game. Combinatorics, Probability and Computing, 25(1), 76–88. https://doi.org/10.1017/S096354831500019X

Clemens, D., Gebauer, H. and Liebenau, A. (2015) ‘The random graph intuition for the tournament game’, Combinatorics, Probability and Computing, 25(1), pp. 76–88. Available at: https://doi.org/10.1017/S096354831500019X.

D. Clemens, H. Gebauer, and A. Liebenau, “The random graph intuition for the tournament game,” Combinatorics, Probability and Computing, vol. 25, no. 1, pp. 76–88, 2015, doi: 10.1017/S096354831500019X.

CLEMENS, Dennis, Heidi GEBAUER und Anita LIEBENAU, 2015. The random graph intuition for the tournament game. Combinatorics, Probability and Computing. 2015. Bd. 25, Nr. 1, S. 76–88. DOI 10.1017/S096354831500019X

Clemens, Dennis, Heidi Gebauer, and Anita Liebenau. 2015. “The Random Graph Intuition for the Tournament Game.” Combinatorics, Probability and Computing 25 (1): 76–88. https://doi.org/10.1017/S096354831500019X.

Clemens, Dennis, et al. “The Random Graph Intuition for the Tournament Game.” Combinatorics, Probability and Computing, vol. 25, no. 1, 2015, pp. 76–88, https://doi.org/10.1017/S096354831500019X.