Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T01:59:53.086Z Has data issue: false hasContentIssue false
  • English
  • Français

Component Games on Regular Graphs

Published online by Cambridge University Press:  04 November 2013

RANI HOD  and
ALON NAOR
RANI HOD
Affiliation:
School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: rani.hod@cs.tau.ac.il)
ALON NAOR
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: alonnaor@tau.ac.il)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the (1:b) Maker–Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players.

To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.

Keywords

Primary 91A24Secondary 05C57
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 75 - 89
Copyright
Copyright © Cambridge University Press 2013 

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]Alon, N. (1997) On the edge-expansion of graphs. Combin. Probab. Comput. 6 145–152.CrossRefGoogle Scholar
[2]Alon, N., Benjamini, I. and Stacey, A. (2004) Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 1727–1745.CrossRefGoogle Scholar
[3]Bednarska, M. and Łuczak, T. (2001) Biased positional games and the phase transition. Random Struct. Alg. 18 141–152.Google Scholar
[4]Bollobás, B. (1978) Chromatic number, girth and maximal degree. Discrete Math. 24 311–314.CrossRefGoogle Scholar
[5]Chvátal, V. and Erdős, P. (1978) Biased positional games. Ann. Discrete Math. 2 221–228.CrossRefGoogle Scholar
[6]Erdős, P. and Sachs, H. (1963) Regular graphs with given girth and minimal number of knots (in German). Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss 12 251–257.Google Scholar
[7]Ferber, A., Glebov, R., Krivelevich, M. and Naor, A. Biased games on random boards. Random Struct. Alg., to appear.Google Scholar
[8]Gallai, T. (1968) On directed graphs and circuits. In Theory of Graphs: Proc. Colloq. Tihany 1966, New York Academic Press, pp. 115–118.Google Scholar
[9]Gebauer, H. and Szabó, T. (2009) Asymptotic random graph intuition for the biased connectivity game. Random Struct. Alg. 35 431–443.CrossRefGoogle Scholar
[10]Hasse, M. (1965) Zur algebraischen Begründung der Graphentheorie I (in German). Mathematische Nachrichten 28 275–290.CrossRefGoogle Scholar
[11]Hefetz, D., Krivelevich, M., Stojaković, M. and Szabó, T. (2011) Global Maker–Breaker games on sparse graphs. Europ. J. Combin. 32 162–177.Google Scholar
[12]Hefetz, D., Mikalački, M. and Stojaković, M. (2012) Doubly biased Maker-Breaker Connectivity game. Electron. J. Combin. 19 P61.Google Scholar
[13]Hoory, S., Linial, N. and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. 43 439–561.Google Scholar
[14]Hod, R., Krivelevich, M., Müller, T. and Naor, A. Component games on random graphs. In preparation.Google Scholar
[15]Lehman, A. (1964) A solution of the Shannon switching game. J. Soc. Indust. Appl. Math. 12 687–725.CrossRefGoogle Scholar
[16]Lubotzky, A., Phillips, R. and Sarnak, P. (1988) Ramanujan graphs. Combinatorica 8 261–277.Google Scholar
[17]Nachmias, A. and Peres, Y. (2010) Critical percolation on random regular graphs. Random Struct. Alg. 36 111–148.Google Scholar
[18]Nash–Williams, C. St J. A. (1961) Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36 445–450.Google Scholar
[19]Roy, B. (1967) Nombre chromatique et plus longs chemins d'un graphe (in French). Rev. Française Informat. Recherche Opérationnelle 1 129–132.Google Scholar
[20]Stojaković, M. and Szabó, T. (2005) Positional games on random graphs. Random Struct. Alg. 26 204–223.Google Scholar
[21]Tutte, W. T. (1961) On the problem of decomposing a graph into n connected factors. J. London Math. Soc. 36 221–230.Google Scholar
[22]Vitaver, L. M. (1962) Determination of minimal colouring of vertices of a graph by means of Boolean powers of the incidence matrix (in Russian). Dokl. Akad. Nauk SSSR 147 758–759.Google Scholar