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Upper Bounds for Online Ramsey Games in Random Graphs

Published online by Cambridge University Press:  01 March 2009

MARTIN MARCINISZYN
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)
RETO SPÖHEL
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)

Abstract

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

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Paper
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Friedgut, E., Kohayakawa, Y., Rödl, V., Ruciński, A. and Tetali, P. (2003) Ramsey games against a one-armed bandit. Combin. Probab. Comput. 12 515545.CrossRefGoogle Scholar
[2]Łuczak, T., Ruciński, A. and Voigt, B. (1992) Ramsey properties of random graphs. J. Combin. Theory Ser. B 56 5568.CrossRefGoogle Scholar
[3]Marciniszyn, M. and Spöhel, R. (2007) Online vertex colorings of random graphs without monochromatic subgraphs. In Proc. 18th annual ACM–SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 486–493. Journal version to appear in Combinatorica.Google Scholar
[4]Marciniszyn, M., Spöhel, R. and Steger, A. (2005) The online clique avoidance game on random graphs. In Approximation, Randomization and Combinatorial Optimization, Vol. 3624 of Lecture Notes in Computer Science, Springer, pp. 390401.Google Scholar
[5]Marciniszyn, M., Spöhel, R. and Steger, A. (2009) Online Ramsey games in random graphs. Combin. Probab. Comput., this issue.Google Scholar
[6]Rödl, V. and Ruciński, A. (1993) Lower bounds on probability thresholds for Ramsey properties. In Combinatorics: Paul Erdős is Eighty, Vol. 1 of Bolyai Society Mathematical Studies, pp. 317–346.Google Scholar
[7]Rödl, V. and Ruciński, A. (1995) Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8 917942.CrossRefGoogle Scholar
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