Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T03:08:09.871Z Has data issue: false hasContentIssue false

8 - Robustness of graph properties

Published online by Cambridge University Press:  21 July 2017

Benny Sudakov
Affiliation:
Department of Mathematics
Anders Claesson
Affiliation:
University of Iceland, Reykjavik
Mark Dukes
Affiliation:
University College Dublin
Sergey Kitaev
Affiliation:
University of Strathclyde
David Manlove
Affiliation:
University of Glasgow
Kitty Meeks
Affiliation:
University of Glasgow
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

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] M., Albert, A., Frieze, and B., Reed, Multicoloured Hamilton cycles. Electron. J. Combin. 2, R10.
[2] P., Allen, J., Böttcher, J., Ehrenmüller and A., Taraz, The bandwidth theorem in sparse graphs, preprint.
[3] N., Alon, The maximum number of Hamiltonian paths in tournaments, Combinatorica 10 (1990), 319–324.Google Scholar
[4] N., Alon and G., Gutin, Properly colored Hamilton cycles in edgecolored complete graphs, Random Struct. Algor. 11 (1997), 179–186.Google Scholar
[5] N., Alon and B., Sudakov, Increasing the chromatic number of a random graph, Journal of Combinatorics 1 (2010), 345–356.Google Scholar
[6] J., Balogh, B., Csaba and W., Samotij, Local resilience of almost spanning trees in random graphs, Random Structures and Algorithms 38 (2011), 121–139.Google Scholar
[7] J., Balogh, C., Lee and W., Samotij, Corrádi and Hajnal's theorem for sparse random graphs, Combin. Probab. Comput. 21 (2012), 23–55.Google Scholar
[8] J., Beck, Combinatorial Games, Cambridge University Press, Cambridge (2008).
[9] S., Ben-Shimon, A., Ferber, D., Hefetz and M., Krivelevich, Hitting time results for Maker-Breaker games, Random Structures and Algorithms 41 (2012), 23–46.Google Scholar
[10] S., Ben-Shimon, M., Krivelevich and B., Sudakov, On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs, SIAM J. Discrete Math. 25 (2011), 1176–1193.Google Scholar
[11] S., Ben-Shimon, M., Krivelevich and B., Sudakov, Local resilience and Hamiltonicity Maker-Breaker games in random regular graphs, Combinatorics, Probability and Computing 20 (2011), 173–211.Google Scholar
[12] B., Bollobás, The evolution of sparse graphs, in Graph Theory and Combinatorics, Academic Press, New York (1984), 35–57.
[13] B., Bollobás, The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.Google Scholar
[14] B., Bollobás, Random Graphs, 2nd ed., Cambridge University Press, Cambridge (2001).
[15] B., Bollobás and P., Erdös, Alternating hamiltonian cycles, Israel J. Math. 23 (1976), 126–131.Google Scholar
[16] B., Bollobás and A., Frieze, On matchings and Hamiltonian cycles in random graphs, in: Random graphs ‘83, North-Holland Math. Stud. 118, North-Holland, Amsterdam, 1985, 23–46.
[17] J., Bondy, Basic Graph Theory: Paths and Circuits, in: Handbook of Combinatorics Vol. 1 (ed. by R., Graham, M., Grötschel and L., Lovász), North-Holland, Amsterdam, 1995, 3–110.
[18] J., Böttcher, Y., Kohayakawa and A., Taraz, Almost spanning subgraphs of random graphs after adversarial edge removal, Combinatorics, Probability and Computing 22 (2013), 639–683.Google Scholar
[19] L. M., Brégman, Some properties of non-negative matrices and their permanents, Sov. Mat. Dokl. 14 (1973), 945–949.Google Scholar
[20] C., Chen and D., Daykin, Graphs with Hamiltonian cycles having adjacent lines different colors, J. Combin. Theory Ser. B 21 (1976), 135–139.Google Scholar
[21] V., Chvátal and P., Erdʺos, Biased positional games, Annals of Discrete Math. 2 (1978), 221–228.Google Scholar
[22] D., Conlon and T., Gowers, Combinatorial theorems in sparse random sets, Annals of Math. 184 (2016), 367–454.Google Scholar
[23] B., Csaba, D., Kühn, A., Lo, D., Osthus and A., Treglown, Proof of the 1-factorization and Hamilton decomposition conjectures, Memoirs of the American Mathematical Society 244 (2016), monograph 1154, 170 pages.Google Scholar
[24] B., Cuckler, Hamilton cycles in regular tournaments, Combinatorics, Probability and Computing 16 (2007), 239–249.Google Scholar
[25] B., Cuckler and J., Kahn, Hamiltonian cycles in Dirac graphs, Combinatorica 29 (2009), 299–326.Google Scholar
[26] D. E., Daykin, Graphs with cycles having adjacent lines different colors, J. Combin. Theory Ser. B 20 (1976), 149–152.Google Scholar
[27] R., Diestel, Graph theory, 3rd ed., Springer-Verlag, Berlin, 2005.
[28] D., Dellamonica, Y., Kohayakawa, M., Marciniszyn and A., Steger, On the resilience of long cycles in random graphs, Electronic J. Combin. 15 (2008), R32.Google Scholar
[29] G., Egorychev, The solution of the Van der Waerden problem for permanents, Dokl. Akad. Nauk SSSR 258 (1981), 1041–1044.Google Scholar
[30] P., Erdʺos, J., Nešsetřril and V., Rödl, On some problems related to partitions of edges of a graph, in: Graphs and other combinatorial topics (Prague, 1982), Teubner-Texte Math. 59, Teubner, Leipzig, 1983, 54– 63.
[31] P., Erdʺos and A., Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.Google Scholar
[32] P., Erdʺos and A., Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14 (1963), 295–315.Google Scholar
[33] P., Erdʺos and A., Rényi, On the existence of a factor of degree one of a connected random graph, Acta Math. Acad. Sci. Hungar. 17 (1966), 359–368.Google Scholar
[34] P., Erdʺos, M., Simonovits and V., Sós, Anti-Ramsey theorems, in: Infinite and finite sets, Vol. 11, edited by A., Hajnal, R, Rado and V., Sós, Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam (1975), 633–643.
[35] D., Falikman, A proof of the Van derWaerden problem for permanents of a doubly stochastic matrix, Mat. Zametki 29 (1981), 931–938.Google Scholar
[36] A., Ferber, R., Glebov, M., Krivelevich and A., Naor, Biased games on random boards, Random Structures and Algorithms 46 (2015), 651– 676.Google Scholar
[37] A., Ferber, M., Krivelevich and H., Naves, Generating random graphs in biased maker-breaker games Random Structures Algorithms 47 (2015), 615–634.Google Scholar
[38] A., Ferber, M., Krivelevich and B., Sudakov, Counting and packing Hamilton cycles in dense graphs and oriented graphs, J. Combinatorial Theory Ser. B, to appear.
[39] A., Ferber, G., Kronenberg and E., Long, Packing, Counting and Covering Hamilton cycles in random directed graphs, Israel J. Mathematics, to appear.
[40] A., Ferber, E., Long and B., Sudakov, Counting Hamiltonian decompositions of oriented graphs, arXiv:1609.09550.
[41] A., Ferber, R., Nenadov, A., Noever, U., Peter and N., Škorić, Robust hamiltonicity of random directed graphs, Proceedings of 26th ACMSIAM SODA, ACM Press (2015), 1752–1758.
[42] A., Frieze and M., Karoński, Introduction to Random Graphs, Cambridge University Press, 2016.
[43] A., Frieze and M., Krivelevich, On packing Hamilton cycles in ε-regular graphs, J. Combinatorial Theory Ser. B 94 (2005), 159–172.Google Scholar
[44] A., Frieze and M., Krivelevich, On two Hamilton cycle problems in random graphs, Israel J. of Mathematics 166 (2008), 221–234.Google Scholar
[45] R., Glebov and M., Krivelevich, On the number of Hamilton cycles in sparse random graphs, SIAM Journal on Discrete Math. 27 (2013), 27–42.Google Scholar
[46] R., Glebov, Z., Luria and B., Sudakov, The number of Hamiltonian decompositions of regular graphs, Israel J. of Mathematics, to appear.
[47] R., Glebov, H., Naves and B., Sudakov, The threshold probability for long cycles, Combinatorics, Probability and Computing, to appear.
[48] R., Häggkvist, Hamilton cycles in oriented graphs, Combinatorics, Probability and Computing 2 (1993), 25–32.Google Scholar
[49] D., Hefetz, M., Krivelevich, M., Stojakovic and T., Szabó, Positional Games, Birkhäuser Basel, 2014.
[50] D., Hefetz, A., Steger and B., Sudakov, Random directed graphs are robustly Hamiltonian, Random Structures and Algorithms 49 (2016), 345–362.Google Scholar
[51] H., Huang, C., Lee and B., Sudakov, Bandwidth theorem for random graphs, J. Combin. Theory Ser. B 102 (2012), 14–37.Google Scholar
[52] S., Janson, T., Łuczak and A., Ruciński, Random graphs, Wiley, New York, 2000.
[53] R., Karp, Reducibility among combinatorial problems, in: Complexity of Computer Computations, New York: Plenum (1972), 85–103.
[54] P., Katerinis, Minimum degree of a graph and the existence of kfactors, Proc. Indian Acad. Sci. Math. Sci. 94 (1985), 123–127.Google Scholar
[55] P., Keevash, D., Kühn and D., Osthus, An exact minimum degree condition for Hamilton cycles in oriented graphs, J. London Math. Soc. 79 (2009), 144–166.Google Scholar
[56] J.H., Kim, B., Sudakov and V., Vu, On asymmetry of random graphs and random regular graphs, Random Structures and Algorithms 21 (2002), 216–224.Google Scholar
[57] F., Knox, D., Kühn and D., Osthus, Edge-disjoint Hamilton cycles in random graphs, Random Structures and Algorithms 46 (2015), 397– 445.Google Scholar
[58] J., Komlós and M., Simonovits, Szemerédi's regularity lemma and its applications in graph theory, in Combinatorics, Paul Erdʺos is eighty, Vol. 2, Bolyai Soc. Math. Stud. 2, Budapest, 1996, 295–352.
[59] J., Komlós and E., Szemerédi, Limit distribution for the existence of Hamiltonian cycles in a random graph, Discrete Mathematics 43 (1983), 55–63.Google Scholar
[60] A., Korshunov, Solution of a problem of Erdʺos and Rényi on Hamilton cycles in non-oriented graphs, Soviet Math. Dokl, 17 (1976), 760–764.Google Scholar
[61] A., Kotzig, Moves without forbidden transitions in a graph, Matematickỳ Časopis 18 (1968), 76–80.Google Scholar
[62] M., Krivelevich, The critical bias for the Hamiltonicity game is (1 + o(1))n/ ln n, Journal of the American Mathematical Society 24 (2011), 125–131.Google Scholar
[63] M., Krivelevich, C., Lee and B., Sudakov, Resilient pancyclicity of random and pseudorandom graphs, SIAM J. Discrete Math. 24 (2010), 1–16.Google Scholar
[64] M., Krivelevich, C., Lee and B., Sudakov, Robust Hamiltonicity of Dirac graphs, Transactions of the Amer. Math. Soc. 366 (2014), 3095–3130.Google Scholar
[65] M., Krivelevich, C., Lee and B., Sudakov, Long paths and cycles in random subgraphs of graphs with large minimum degree, Random Structures and Algorithms 46 (2015), 320–345.Google Scholar
[66] M., Krivelevich, C., Lee and B., Sudakov, Compatible Hamilton cycles in random graphs, Random Structures and Algorithms 49 (2016), 533– 557.Google Scholar
[67] M., Krivelevich, C., Lee and B., Sudakov, Compatible Hamilton cycles in Dirac graphs, Combinatorica, to appear.
[68] M., Krivelevich and W., Samotij, Optimal packings of Hamilton cycles in sparse random graphs, SIAM J. Discrete Mathematics 26 (2012), 964–982.Google Scholar
[69] M., Krivelevich and B., Sudakov, Sparse pseudo-random graphs are Hamiltonian, J. Graph Theory 42 (2003), 17–33.Google Scholar
[70] M., Krivelevich and B., Sudakov, Pseudo-random graphs, in: More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer, 2006, 199–262.
[71] M., Krivelevich, B., Sudakov, V., Vu and N., Wormald, Random regular graphs of high degree, Random Structures and Algorithms 18 (2001), 346–363.Google Scholar
[72] M., Krivelevich, B., Sudakov, V., Vu and N., Wormald, On the probability of independent sets in random graphs, Random Structures and Algorithms 22 (2003), 1–14.Google Scholar
[73] D., Kühn, J., Lapinskas and D., Osthus, Optimal packings of Hamilton cycles in graphs of high minimum degree, Combinatorics, Probability, Computing 22 (2013), 394–416.Google Scholar
[74] D., Kühn and D., Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62–146.Google Scholar
[75] D., Kühn and D., Osthus, Hamilton decompositions of regular expanders: applications, J. Combinatorial Theory Series B 104 (2014), 1–27.Google Scholar
[76] D., Kühn, D., Osthus and A., Treglown, Hamilton decompositions of regular tournaments, Proceedings of the London Mathematical Society 101 (2010), 303–335.Google Scholar
[77] C., Lee and B., Sudakov, Dirac's theorem for random graphs, Random Structures and Algorithms 41 (2012), 293–305.Google Scholar
[78] A., Lo, Properly coloured Hamiltonian cycles in edge-coloured complete graphs, Combinatorica, to appear.
[79] L., Lovász, Combinatorial problems and exercises, 2nd edition, AMS Chelsea Publishing, 2007.
[80] T., Łuczak, The chromatic number of random graphs, Combinatorica 11 (1991), 45–54.Google Scholar
[81] C., Nash-Williams, Hamiltonian lines in graphs whose vertices have sufficiently large valencies, in: Combinatorial Theory and Its Applications, III, North-Holland, 1970, 813–819.
[82] C., Nash-Williams, Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency, in: Studies in Pure Mathematics, Academic Press, London, 1971, 157–183.
[83] A., Noever and A., Steger, Local resilience for squares of almost spanning cycles in sparse random graphs, preprint.
[84] L., Pósa, Hamiltonian circuits in random graphs, Discrete Mathematics, 14 (1976), 359–364.Google Scholar
[85] R., Rado, Anti-Ramsey theorems, in: Infinite and finite sets, Vol. 11, edited by A., Hajnal, R., Rado and V., Sós, Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam (1975), 1159–1168.
[86] O., Riordan, Long cycles in random subgraphs of graphs with large minimum degree Random Structures and Algorithms 45 (2014), 764– 767.Google Scholar
[87] R., Robinson, and N., Wormald, Almost all regular graphs are Hamiltonian Random Structures and Algorithms 5 (1994), 363–374.Google Scholar
[88] G., Sárközy, S., Selkow and E., Szemerédi, On the number of Hamiltonian cycles in Dirac graphs, Discrete Math. 265 (2003), 237–250.Google Scholar
[89] M., Schacht, Extremal results for random discrete structures, Annals of Math. 184 (2016), 333–365.Google Scholar
[90] B., Sudakov and V., Vu, Local resilience of graphs, Random Structures and Algorithms 33 (2008), 409–433.Google Scholar
[91] T., Szele, Kombinatorikai vizsgalatok az iranyitott teljes graffal, Mt. Fiz. Lapok 50 (1943), 223–256.Google Scholar
[92] C., Thomassen, Long cycles in digraphs with constraints on the degrees, in: Surveys in Combinatorics (B., Bollobás ed.), London Math. Soc. Lecture Notes 38, Cambridge University Press, 1979, 211–228.
[93] C., Thomassen, Hamilton circuits in regular tournaments, Annals of Discrete Math. 27 (1985), 159–162.Google Scholar
[94] N.C., Wormald, Models of random regular graphs, in: Surveys in Combinatorics, Cambridge University Press, 1999, 239–298.

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×