Abstract
We study Maker-Breaker k-clique game played on the edge set of the random graph G(n, p). In this game, two players alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win is at \( n^{ - \frac{2} {{k + 1}}} \), for all k > 3, thus proving a conjecture from [5]. More precisely, we conclude that there exist constants c, C > 0 such that when \( p < Cn^{ - \frac{2} {{k + 1}}} \) the game is Maker’s win a.a.s., and when \( p < cn^{ - \frac{2} {{k + 1}}} \) it is Breaker’s win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker’s win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K 5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker’s win in the triangle game played on the edge set of G(n, p).
The first author was supported in part by a VENI grant from Netherlands Organization for Scientific Research. The second author was partly supported by Ministry of Education and Science, Republic of Serbia, and Provincial Secretariat for Science, Province of Vojvodina.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Bednarska and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica 20 (2000), 477–488.
S. Ben-Shimon, A. Ferber, D. Hefetz and M. Krivelevich, Hitting time results for Maker-Breaker games, Random Structures and Algorithms 41 (2012), 23–46.
D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó, A sharp threshold for the Hamilton cycle Maker-Breaker game, Random Structures and Algorithms 34 (2009), 112–122.
M. Stojaković, “Games on Graphs”, PhD Thesis, ETH Zürich, 2005.
M. Stojaković and T. Szabö, Positional games on random graphs, Random Structures and Algorithms 26 (2005), 204–223.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Müller, T., Stojaković, M. (2013). A threshold for the Maker-Breaker clique game. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_56
Download citation
DOI: https://doi.org/10.1007/978-88-7642-475-5_56
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-474-8
Online ISBN: 978-88-7642-475-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)