Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-mm7gn Total loading time: 0.36 Render date: 2022-08-11T21:49:13.876Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

The Random Graph Intuition for the Tournament Game

Published online by Cambridge University Press:  23 September 2015

DENNIS CLEMENS
Affiliation:
Institut für Mathematik, Technische Universität Hamburg–Harburg, Germany (e-mail: dennis.clemens@tuhh.de)
HEIDI GEBAUER
Affiliation:
Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Switzerland (e-mail: geba@zhaw.ch)
ANITA LIEBENAU
Affiliation:
School of Mathematical Sciences, Monash University VIC 3800, Australia (e-mail: anita.liebenau@monash.edu)

Abstract

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, 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))log2n Breaker can prevent the underlying graph of Maker's digraph from containing 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 of 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 – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

Type
Paper
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Balogh, R., Martin, J. and Pluhár, A. (2009) The diameter game. Random Struct. Alg. 35 369389.CrossRefGoogle Scholar
[2]Beck, J. (1985) Random graphs and positional games on the complete graph. Ann. Discrete Math. 28 713.Google Scholar
[3]Beck, J. (1993) Achievement games and the probabilistic method. In Combinatorics: Paul Erdős is Eighty, Vol. 1, János Bolyai Mathematical Society, pp. 51–78.Google Scholar
[4]Beck, J. (1994) Deterministic graphs games and a probabilistic intuition. Combin. Probab. Comput. 3 1326.CrossRefGoogle Scholar
[5]Beck, J. (1996) Foundations of positional games. Random Struct. Alg. 9 1547.3.0.CO;2-E>CrossRefGoogle Scholar
[6]Beck, J. (2008) Combinatorial Games: Tic-Tac-Toe Theory, Vol. 114 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[7]Bednarska, M. and Łuczak, T. (2000) Biased positional games for which random strategies are nearly optimal. Combinatorica 20 477488.CrossRefGoogle Scholar
[8]Bednarska, M. and Łuczak, T. (2001) Biased positional games and the phase transition. Random Struct. Alg. 18 141152.3.0.CO;2-W>CrossRefGoogle Scholar
[9]Ben-Eliezer, I., Krivelevich, M. and Sudakov, B. (2012) Biased orientation games. Discrete Math. 312 17321742.CrossRefGoogle Scholar
[10]Bollobás, B. (2001) Random Graphs, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[11]Bollobás, B. and Szabó, T. (1998) The oriented cycle game. Discrete Math. 186 5567.CrossRefGoogle Scholar
[12]Chartrand, G., Harary, F., Schultz, M. and VanderJagt, D. W. (1995) Achievement and avoidance of a strong orientation of a graph. Congressus Numerantium, pp. 193204.Google Scholar
[13]Chvátal, V. and Erdős, P. (1978) Biased positional games. Ann. Discrete Math. 2 221228.CrossRefGoogle Scholar
[14]Clemens, D. and Liebenau, A. (2014) A non-trivial upper bound on the threshold bias of the Oriented-cycle game. arXiv:1404.4529Google Scholar
[15]Erdős, P. and Selfridge, J. (1973) On a combinatorial game. J. Combin. Theory Ser. A 14 298301.CrossRefGoogle Scholar
[16]Gebauer, H. (2012) On the clique-game. European J. Combin. 33 819.CrossRefGoogle Scholar
[17]Gebauer, H. and Szabó, T. (2009) Asymptotic random graph intuition for the biased connectivity game. Random Struct. Alg. 35 431443.CrossRefGoogle Scholar
[18]Hefetz, D., Krivelevich, M., Stojakovic, M. and Szabó, T. (2008) Planarity, colorability, and minor games. SIAM J. Discrete Math. 22 194212.CrossRefGoogle Scholar
[19]Stojaković, M. and Szabó, T. (2005) Positional games on random graphs. Random Struct. Alg. 26 204223.CrossRefGoogle Scholar
1
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The Random Graph Intuition for the Tournament Game
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

The Random Graph Intuition for the Tournament Game
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

The Random Graph Intuition for the Tournament Game
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *