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Almost Spanning Subgraphs of Random Graphs After Adversarial Edge Removal

Published online by Cambridge University Press:  08 August 2013

JULIA BÖTTCHER ,
YOSHIHARU KOHAYAKAWA  and
ANUSCH TARAZ
JULIA BÖTTCHER
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508–090 São Paulo, Brazil (e-mail: boettche@lse.ac.uk)
YOSHIHARU KOHAYAKAWA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508–090 São Paulo, Brazil (e-mail: yoshi@ime.usp.br)
ANUSCH TARAZ
Affiliation:
Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D–85747 Garching bei München, Germany (e-mail: taraz@ma.tum.de)
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Abstract

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Let Δ ≥ 2 be a fixed integer. We show that the random graph ${\mathcal{G}_{n,p}}$ with $p\gg (\log n/n)^{1/\Delta}$ is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from ${\mathcal{G}_{n,p}}$ in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.

Keywords

Primary 05C35Secondary 05C8005C60

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 5 , September 2013 , pp. 639 - 683
Copyright
Copyright © Cambridge University Press 2013 

References

[1]Albert, M., Frieze, A. and Reed, B. (1995) Multicoloured Hamilton cycles. Electron. J. Combin. 2 13 pp.CrossRefGoogle Scholar
[2]Alon, N. and Capalbo, M. (2007) Sparse universal graphs for bounded-degree graphs. Random Struct. Alg. 31 123–133.CrossRefGoogle Scholar
[3]Alon, N. and Capalbo, M. (2008) Optimal universal graphs with deterministic embedding. In Proc. 19th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA, pp. 373–378.Google Scholar
[4]Alon, N., Capalbo, M., Kohayakawa, Y., Rödl, V., Ruciński, A. and Szemerédi, E. (2000) Universality and tolerance. In Proc. 41st Annual Symposium on Foundations of Computer Science: FOCS, pp. 14–21.CrossRefGoogle Scholar
[5]Alon, N., Krivelevich, M. and Sudakov, B. (2007) Embedding nearly-spanning bounded degree trees. Combinatorica 27 629–644.Google Scholar
[6]Balogh, J., Csaba, B. and Samotij, W. (2011) Local resilience of almost spanning trees in random graphs. Random Struct. Alg. 38 121–139.Google Scholar
[7]Balogh, J., Lee, C. and Samotij, W. (2012) Corrádi and Hajnal's theorem for sparse random graphs. Combin. Probab. Comput. 21 23–55.Google Scholar
[8]Böttcher, J., Kohayakawa, Y. and Taraz, A. Almost spanning subgraphs of random graphs after adversarial edge removal. arXiv:1003.0890Google Scholar
[9]Böttcher, J., Kohayakawa, Y. and Procacci, A. (2012) Properly coloured copies and rainbow copies of large graphs with small maximum degree. Random Struct. Alg. 40 425–436.CrossRefGoogle Scholar
[10]Böttcher, J., Pruessmann, K. P., Taraz, A. and Würfl, A. (2010) Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. Europ. J. Combin. 31 1217–1227.CrossRefGoogle Scholar
[11]Böttcher, J., Schacht, M. and Taraz, A. (2009) Proof of the bandwidth conjecture of Bollobás and Komlós. Math. Ann. 343 175–205.CrossRefGoogle Scholar
[12]Dellamonica, D., Kohayakawa, Y., Marciniszyn, M. and Steger, A. (2008) On the resilience of long cycles in random graphs. Electron. J. Combin. 15 R32.Google Scholar
[13]Dellamonica, D., Kohayakawa, Y., Rödl, V. and Ruciński, A. (2008) Universality of random graphs. In Proc. 19th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA, pp. 782–788.Google Scholar
[14]Dellamonica, D. Jr, Kohayakawa, Y., Rödl, V. and Ruciński, A. (2012) An improved upper bound on the density of universal random graphs. In LATIN 2012, Vol. 7256 of Lecture Notes in Computer Science, Springer, pp. 231–242.Google Scholar
[15]Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. (3) 2 69–81.CrossRefGoogle Scholar
[16]Erdős, P., Nešetřil, J. and Rödl, V. (1983) On some problems related to partitions of edges of a graph. In Graphs and Other Combinatorial Topics: Prague 1982, Teubner, pp. 54–63.Google Scholar
[17]Frieze, A. and Reed, B. (1993) Polychromatic Hamilton cycles. Discrete Math. 118 69–74.Google Scholar
[18]Gerke, S., Kohayakawa, Y., Rödl, V. and Steger, A. (2007) Small subsets inherit sparse ε-regularity. J. Combin. Theory Ser. B 97 34–56.Google Scholar
[19]Hahn, G. and Thomassen, C. (1986) Path and cycle sub-Ramsey numbers and an edge-colouring conjecture. Discrete Math. 62 29–33.Google Scholar
[20]Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications II: Balatonfüred 1969, North-Holland, pp. 601–623.Google Scholar
[21]Huang, H., Lee, C. and Sudakov, B. (2012) Bandwidth theorem for random graphs. J. Combin. Theory Ser. B 102 14–37.Google Scholar
[22]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience.Google Scholar
[23]Kohayakawa, Y. (1997) Szemerédi's regularity lemma for sparse graphs. In Foundations of Computational Mathematics: Rio de Janeiro 1997, Springer, pp. 216–230.Google Scholar
[24]Kohayakawa, Y. and Rödl, V. (2003) Regular pairs in sparse random graphs I. Random Struct. Alg. 22 359–434.CrossRefGoogle Scholar
[25]Kohayakawa, Y. and Rödl, V. (2003) Szemerédi's regularity lemma and quasi-randomness. In Recent Advances in Algorithms and Combinatorics, Vol. 11 of CMS Books Math./Ouvrages Math. SMC., Springer, pp. 289–35.Google Scholar
[26]Kohayakawa, Y., Rödl, V., Schacht, M. and Szemerédi, E. (2011) Sparse partition universal graphs for graphs of bounded degree. Adv. Math. 226 5041–5065.CrossRefGoogle Scholar
[27]Rödl, V., Ruciński, A. and Taraz, A. (1999) Hypergraph packing and graph embedding. Combin. Probab. Comput. 8 363–376.CrossRefGoogle Scholar
[28]Sudakov, B. and Vu, V. (2008) Local resilience of graphs. Random Struct. Alg. 33 409–433.CrossRefGoogle Scholar
[29]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 436–452.Google Scholar