Hostname: page-component-5db58dd55d-mhzq2 Total loading time: 0 Render date: 2026-05-25T20:42:41.066Z Has data issue: false hasContentIssue false
  • English
  • Français

Maker–Breaker percolation games I: crossing grids

Part of: Extremal combinatorics Game theory Graph theory

Published online by Cambridge University Press:  15 September 2020

A. Nicholas Day  and
Victor Falgas-Ravry
A. Nicholas Day
Affiliation:
Institutionen för Matematik och Matematisk Statistik, Umeå Universitet, 901 87 Umeå, Sweden.
Victor Falgas-Ravry*
Affiliation:
Institutionen för Matematik och Matematisk Statistik, Umeå Universitet, 901 87 Umeå, Sweden.
*
*Corresponding author. Email: victor.falgas-ravry@umu.se
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n.

Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition.

  • If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ.

  • If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m.

Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

MSC classification

Primary: 05C57: Games on graphs
Secondary: 05D99: None of the above, but in this section 91A43: Games involving graphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 2 , March 2021 , pp. 200 - 227
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Footnotes

†

Research supported by Swedish Research Council grant 2016-03488.

References

Beck, J. (1982) Remarks on positional games I. Acta Math. Hungar. 40 65–71.CrossRefGoogle Scholar
Beck, J. (1985) Random graphs and positional games on the complete graph. In Random Graphs ’83, Vol. 118 of North-Holland Mathematics Studies, pp. 7–13, Elsevier.Google Scholar
Beck, J. (1993) Achievement games and the probabilistic method. In Combinatorics: Paul Erdös is Eighty, Vol. 1, pp. 51–78, János Bolyai Mathematical Society.Google Scholar
Beck, J. (1994) Deterministic graph games and a probabilistic intuition. Combin. Probab. Comput. 3 13–26.CrossRefGoogle Scholar
Beck, J. (2008) Combinatorial Games: Tic-Tac-Toe Theory, Vol. 114 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
Bednarska, M. and Łuczak, T. (2000) Biased positional games for which random strategies are nearly optimal. Combinatorica 20 477–488.CrossRefGoogle Scholar
Bednarska, M. and Łuczak, T. (2001) Biased positional games and the phase transition. Random Struct. Algorithms 18 141–152.3.0.CO;2-W>CrossRefGoogle Scholar
Bollobás, B. and Riordan, O. (2006) Percolation, Cambridge University Press.CrossRefGoogle Scholar
Bousquet-Mélou, M. Guttmann, A. J. and Jensen, I. (2005) Self-avoiding walks crossing a square. J. Phys. A Math. Gen. 38 9159.CrossRefGoogle Scholar
Chvátal, V. and Erdös, P. (1978) Biased positional games. Ann. Discrete Math. 2 221–229.CrossRefGoogle Scholar
Day, A. N. and Falgas-Ravry, V. (2020) Maker–Breaker percolation games II: Escaping to infinity. Journal of Combinatorial Theory, Series B. doi: https://doi.org/10.1016/j.jctb.2020.06.006 Google Scholar
Erdös, P. and Selfridge, J. L. (1973) On a combinatorial game. J. Combin. Theory Ser. A 14 298–301.CrossRefGoogle Scholar
Gebauer, H. and Szabó, T. (2009) Asymptotic random graph intuition for the biased connectivity game. Random Struct. Algorithms 35 431–443.CrossRefGoogle Scholar
Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Math. Proc. Camb. Phil. Soc. 56 13–20.CrossRefGoogle Scholar
Hayward, R. B. and van Rijswijck, J. (2006) Hex and combinatorics. Discrete Math. 306 2515–2528.CrossRefGoogle Scholar
Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74 41–59.CrossRefGoogle Scholar
Krivelevich, M. (2011) The critical bias for the Hamiltonicity game is (1+o(1))n/ln n. J. Amer. Math. Soc. 24 125–131.CrossRefGoogle Scholar
Kusch, C., Rué, J., Spiegel, C. and Szabó, T. (2019) On the optimality of the uniform random strategy. Random Struct. Algorithms 55 371–401.CrossRefGoogle Scholar
Lehman, A. (1964) A solution of the Shannon switching game. J. Soc. Indust. Appl. Math. 12 687–725.CrossRefGoogle Scholar
Mansfield, R. (1996) Strategies for the Shannon switching game. Amer. Math. Monthly 103 250–252.CrossRefGoogle Scholar
Stojaković, M. and Szabó, T. (2005) Positional games on random graphs. Random Struct. Algorithms 26 204–223.CrossRefGoogle Scholar