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Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs

Published online by Cambridge University Press:  16 December 2010

SONNY BEN-SHIMON ,
MICHAEL KRIVELEVICH  and
BENNY SUDAKOV
SONNY BEN-SHIMON
Affiliation:
School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: sonny@tau.ac.il)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: krivelev@tau.ac.il)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, UCLA, Los Angles 90005, CA, USA (e-mail: bsudakov@math.ucla.edu)
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Abstract

For an increasing monotone graph property the local resilience of a graph G with respect to is the minimal r for which there exists a subgraph H ⊆ G with all degrees at most r, such that the removal of the edges of H from G creates a graph that does not possess . This notion, which was implicitly studied for some ad hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model (n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random regular graphs of constant degree. We investigate the local resilience of the typical random d-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values of d, with high probability, the local resilience of the random d-regular graph, n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model (n, p), for every positive ϵ > 0 and large enough values of K, if p > then, with high probability, the local resilience of (n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that if d is large enough then, with high probability, a typical random d-regular graph G is such that, in the unbiased Maker–Breaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 173 - 211
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
[2]Beck, J. (2008) Combinatorial Games: Tic-Tac-Toe Theory, Cambridge University Press.CrossRefGoogle Scholar
[3]Bollobás, B. (1981) Random graphs. In Combinatorics (Temperley, H. N. V., ed.), Vol. 52 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 80–102.CrossRefGoogle Scholar
[4]Bollobás, B. (1983) Almost all regular graphs are Hamiltonian. Europ. J. Combin. 4 94–106.CrossRefGoogle Scholar
[5]Bollobás, B. (2001) Random Graphs, Cambridge University Press.CrossRefGoogle Scholar
[6]Broder, A., Frieze, A., Suen, S. and Upfal, E. (1999) Optimal construction of edge-disjoint paths in random graphs. SIAM J. Comput. 28 541–573.CrossRefGoogle Scholar
[7]Chung, F. (2004) Discrete isoperimetric inequalities. In Surveys in Differential Geometry: Eigenvalues of Laplacians and Other Geometric Operators (Grigor'yan, A. and Yau, S. T., eds), Vol. IX, International Press.Google Scholar
[8]Chvátal, V. and Erdős, P. (1978) Biased positional games. Ann. Discrete Math. 2 221–228.CrossRefGoogle Scholar
[9]Dellamonica, D., Kohayakawa, Y., Marciniszyn, M. and Steger, A. (2008) On the resilience of long cycles in random graphs. Electron. J. Combin. 15 R32.CrossRefGoogle Scholar
[10]Diestel, R. (2005) Graph Theory, third edition, Vol. 173 of Graduate Texts in Mathematics, Springer.Google Scholar
[11]Fenner, T. and Frieze, A. (1984) Hamiltonian cycles in random regular graphs. J. Combin. Theory Ser. B 37 103–198.CrossRefGoogle Scholar
[12]Friedman, J. (2008) A proof of Alon's second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195 (910).Google Scholar
[13]Frieze, A. (1988) Finding Hamilton cycles in sparse random graphs. J. Combin. Theory Ser. B 44 230–250.CrossRefGoogle Scholar
[14]Frieze, A. and Krivelevich, M. (2008) On two Hamiltonian cycle problems in random graphs. Israel J. Math. 166 221–234.CrossRefGoogle Scholar
[15]Greenhill, C., Kim, J. H., Janson, S. and Wormald, N. C. (2002) Permutation pseudographs and contiguity. Combin. Probab. Comput. 11 273–298.CrossRefGoogle Scholar
[16]Hefetz, D., Krivelevich, M., Stojaković, M. and Szabó, T. (2009) A sharp threshold for the Hamilton cycle Maker–Breaker game. Random Struct. Alg. 34 112–122.CrossRefGoogle Scholar
[17]Hefetz, D., Krivelevich, M., Stojaković, M. and Szabó, T. Global Maker-Breaker games on sparse graphs. Europ. J. Combin., to appear.Google Scholar
[18]Hefetz, D., Krivelevich, M. and Szabó, T. (2009) Hamilton cycles in highly connected and expanding graphs. Combinatorica 29 547–568.CrossRefGoogle Scholar
[19]Hefetz, D. and Stich, S. (2009) On two problems regarding the Hamilton cycle game. Electron. J. Combin. 16 R28.CrossRefGoogle Scholar
[20]Janson, S. (1995) Random regular graphs: Asymptotic distributions and contiguity. Combin. Probab. Comput. 4 369–405.CrossRefGoogle Scholar
[21]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
[22]Kim, J. H., Sudakov, B. and Vu, V. H. (2002) On the asymmetry of random regular graphs. Random Struct. Alg. 21 216–224.CrossRefGoogle Scholar
[23]Kim, J. H., Sudakov, B. and Vu, V. H. (2007) Small subgraphs of random regular graphs. Discrete Math. 307 1961–1967.CrossRefGoogle Scholar
[24]Kim, J. H. and Vu, V. H. (2004) Sandwiching random graphs. Adv. Math. 188 444–469.CrossRefGoogle Scholar
[25]Kim, J. H. and Wormald, N. C. (2001) Random matchings which induce Hamilton cycles, and Hamiltonian decompositions of random regular graphs. J. Combin. Theory Ser. B 81 20–44.CrossRefGoogle Scholar
[26]Krivelevich, M., Lee, C. and Sudakov, B. (2010) Resilient pancyclicity of random and pseudo-random graphs. SIAM J. Discrete Math. 24 1–16.CrossRefGoogle Scholar
[27]Krivelevich, M. and Sudakov, B. (2003) Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42 17–33.CrossRefGoogle Scholar
[28]Krivelevich, M. and Sudakov, B. (2006) Pseudo-random graphs. In More Sets, Graphs and Numbers: A Salute to Vera Sòs and András Hajnal (Győri, E., Katona, G. O. H. and Lovász, L., eds), Vol. 15 of Bolyai Society Mathematical Studies, Springer, pp. 199–262.CrossRefGoogle Scholar
[29]Krivelevich, M., Sudakov, B., Vu, V. H. and Wormald, N. (2001) Random regular graphs of high degree. Random Struct. Alg. 18 346–363.CrossRefGoogle Scholar
[30]Lehman, A. (1964) A solution of the Shannon switching game. J. Soc. Ind. Appl. Math. 12 687–725.CrossRefGoogle Scholar
[31]McDiarmid, C. (1998) Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics (Habib, M., McDiarmid, C., Ramirez-Alfonsin, J. and Reed, B., eds), Springer, pp. 195–248.CrossRefGoogle Scholar
[32]McKay, B. D. (1985) Asymptotics for symmetric 0–1 matrices with prescribed row sums. Ars Combinatorica 19 15–26.Google Scholar
[33]Nilli, A. (1991) On the second eigenvalue of a graph. Discrete Math. 91 207–210.CrossRefGoogle Scholar
[34]Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.CrossRefGoogle Scholar
[35]Robinson, R. and Wormald, N. (1992) Almost all cubic graphs are Hamiltonian. Random Struct. Alg. 3 117–125.CrossRefGoogle Scholar
[36]Robinson, R. and Wormald, N. (1994) Almost all regular graphs are Hamiltonian. Random Struct. Alg. 5 363–374.CrossRefGoogle Scholar
[37]Sudakov, B. and Vu, V. H. (2008) The local resilience of random graphs. Random Struct. Alg. 33 409–433.CrossRefGoogle Scholar
[38]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 436–452.Google Scholar
[39]Wormald, N. C. (1981) The asymptotic connectivity of labelled regular graphs. J. Combin. Theory Ser. B 31 156–167.CrossRefGoogle Scholar
[40]Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics (Lamb, J. and Preece, D., eds), Vol. 276 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 239–298.Google Scholar