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2 - Large-scale structures in random graphs

[1] R., Aharoni and P., Haxell, Hall's theorem for hypergraphs, J. Graph Theory 35 (2000), no. 2, 83–88.
[2] M., Ajtai, J., Komlós, and E., Szemerédi, The longest path in a random graph, Combinatorica 1 (1981), no. 1, 1–12.
[3] P., Allen, J., Böttcher, J., Ehrenmüller, and A., Taraz, The bandwidth theorem in sparse graphs, arXiv:1612.00661.
[4] P., Allen, J., Böttcher, S., Griffiths, Y., Kohayakawa, and R., Morris, Chromatic thresholds in sparse random graphs, Random Structures Algorithms, accepted, arXiv:1508.03875.
[5] P., Allen, J., Böttcher, H., Hàn, Y., Kohayakawa, and Y., Person, Blowup lemmas for sparse graphs, arXiv:1612.00622.
[6] N., Alon, Universality, tolerance, chaos and order, An irregular mind, Bolyai Soc. Math. Stud., vol. 21, János Bolyai Math. Soc., Budapest, 2010, pp. 21–37.
[7] N., Alon and M., Capalbo, Sparse universal graphs for bounded-degree graphs, Random Structures Algorithms 31 (2007), no. 2, 123–133.
[8] N., Alon and M., Capalbo, Optimal universal graphs with deterministic embedding, Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2008, pp. 373–378.
[9] N., Alon, M., Capalbo, Y., Kohayakawa, V., Rödl, A., Ruciński, and E., Szemerédi, Universality and tolerance (extended abstract), 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, pp. 14–21.
[10] N., Alon and Z., Füredi, Spanning subgraphs of random graphs, Graphs Combin. 8 (1992), no. 1, 91–94.
[11] N., Alon, M., Krivelevich, and B., Sudakov, Embedding nearlyspanning bounded degree trees, Combinatorica 27 (2007), no. 6, 629– 644.
[12] N., Alon and R., Yuster, Threshold functions for H-factors, Combin. Probab. Comput. 2 (1993), no. 2, 137–144.
[13] D., Angluin and L. G., Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comput. System Sci. 18 (1979), no. 2, 155–193.
[14] L., Babai, M., Simonovits, and J., Spencer, Extremal subgraphs of random graphs, J. Graph Theory 14 (1990), no. 5, 599–622.
[15] D., Bal and A., Frieze, The Johansson-Kahn-Vu solution of the Shamir problem, https://www.math.cmu.edu/∼af1p/Teaching/ATIRS/Papers/FRH/Shamir.pdf.
[16] J., Balogh, B., Csaba, M., Pei, and W., Samotij, Large bounded degree trees in expanding graphs, Electron. J. Combin. 17 (2010), no. 1, Research Paper 6, 9.
[17] J., Balogh, B., Csaba, and W., Samotij, Local resilience of almost spanning trees in random graphs, Random Structures Algorithms 38 (2011), no. 1-2, 121–139.
[18] J., Balogh, C., Lee, and W., Samotij, Corrádi and Hajnal's theorem for sparse random graphs, Combin. Probab. Comput. 21 (2012), no. 1-2, 23–55.
[19] J., Balogh, R., Morris, and W., Samotij, Independent sets in hypergraphs, J. Amer. Math. Soc. 28 (2015), no. 3, 669–709.
[20] S., Ben-Shimon, M., Krivelevich, and B., Sudakov, Local resilience and Hamiltonicity maker-breaker games in random regular graphs, Combin. Probab. Comput. 20 (2011), no. 2, 173–211.
[21] 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), no. 3, 1176–1193.
[22] P., Bennett, A., Dudek, and A., Frieze, Square of a Hamilton cycle in a random graph, arXiv:1611.06570.
[23] S. N., Bhatt, F. R. K., Chung, F. T., Leighton, and A. L., Rosenberg, Universal graphs for bounded-degree trees and planar graphs, SIAM J. Discrete Math. 2 (1989), no. 2, 145–155.
[24] B., Bollobás, Threshold functions for small subgraphs, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 197–206.
[25] B., Bollobás, The evolution of sparse graphs, Graph theory and combinatorics (Cambridge, 1983), Academic Press, London, 1984, pp. 35–57.
[26] B., Bollobás, Random graphs, second ed., Cambridge Studies in Advanced Mathematics, vol. 73, Cambridge University Press, Cambridge, 2001.
[27] B., Bollobás, T. I., Fenner, and A. M., Frieze, An algorithm for finding Hamilton paths and cycles in random graphs, Combinatorica 7 (1987), no. 4, 327–341.
[28] B., Bollobás and A. M., Frieze, On matchings and Hamiltonian cycles in random graphs, Ann. Discrete Math. 28 (1985), 23–46.
[29] B., Bollobás and A. M., Frieze, Spanning maximal planar subgraphs of random graphs, Random Structures Algorithms 2 (1991), no. 2, 225–231.
[30] B., Bollobás and A., Thomason, Random graphs of small order, Random graphs ‘83 (Poznań, 1983), North-Holland Math. Stud., vol. 118, North-Holland, Amsterdam, 1985, pp. 47–97.
[31] B., Bollobás and A., Thomason, Threshold functions, Combinatorica 7 (1987), no. 1, 35–38.
[32] J., Böttcher, Y., Kohayakawa, and A., Taraz, Almost spanning subgraphs of random graphs after adversarial edge removal, Combin. Probab. Comput. 22 (2013), no. 5, 639–683.
[33] J., Böttcher, K. P., Pruessmann, A., Taraz, and A., Würfl, Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs, European J. Combin. 31 (2010), no. 5, 1217–1227.
[34] J., Böttcher, M., Schacht, and A., Taraz, Proof of the bandwidth conjecture of Bollobás and Komlós, Math. Ann. 343 (2009), no. 1, 175– 205.
[35] G., Brightwell, K., Panagiotou, and A., Steger, Extremal subgraphs of random graphs, Random Structures Algorithms 41 (2012), no. 2, 147–178.
[36] M. R., Capalbo, A small universal graph for bounded-degree planar graphs, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 1999), ACM, New York, 1999, pp. 156–160.
[37] F. R. K., Chung, Labelings of graphs, Selected topics in graph theory, 3, Academic Press, San Diego, CA, 1988, pp. 151–168.
[38] D., Conlon, A., Ferber, R., Nenadov, and N., Škorić, Almost-spanning universality in random graphs, Random Structures Algorithms, accepted, arXiv:1503.05612.
[39] D., Conlon and W. T., Gowers, Combinatorial theorems in sparse random sets, Ann. of Math. (2) 184 (2016), no. 2, 367–454.
[40] D., Conlon, W. T., Gowers, W., Samotij, and M., Schacht, On the KŁR conjecture in random graphs, Israel J. Math. 203 (2014), no. 1, 535– 580.
[41] K., Corrádi and A., Hajnal, On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423– 439.
[42] D., Dellamonica, Y., Kohayakawa, M., Marciniszyn, and A., Steger, On the resilience of long cycles in random graphs, Electron. J. Combin. 15 (2008), 26 pp., R32.
[43] D., Dellamonica, Y., Kohayakawa, V., Rödl, and A., Ruciński, Universality of random graphs, SIAM J. Discrete Math. 26 (2012), no. 1, 353–374.
[44] D., Dellamonica, Y., Kohayakawa, V., Rödl, and A., Ruciński, An improved upper bound on the density of universal random graphs, Random Structures Algorithms 46 (2015), no. 2, 274– 299.
[45] B., DeMarco and J., Kahn, Turán's Theorem for random graphs, arXiv:1501.01340.
[46] B., DeMarco and J., Kahn, Mantel's theorem for random graphs, Random Structures Algorithms 47 (2015), no. 1, 59–72.
[47] G. A., Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952), 69–81.
[48] M., Drmota, O., Giménez, M., Noy, K., Panagiotou, and A., Steger, The maximum degree of random planar graphs, Proc. Lond. Math. Soc. (3) 109 (2014), no. 4, 892–920.
[49] P., Erdʺos and A., Rényi, On random graphs. I, Publ. Math. Debrecen 6 (1959), 290–297.
[50] P., Erdʺos and A., Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.
[51] P., Erdʺos and A., Rényi, On random matrices, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1964), 455–461 (1964).
[52] 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.
[53] P., Erdʺos and A. H., Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087–1091.
[54] A., Ferber, G., Kronenberg, and K., Luh, Optimal threshold for a random graph to be 2-universal, arXiv:1612.06026.
[55] A., Ferber, K., Luh, and O., Nguyen, Embedding large graphs into a random graph, arXiv:1606.05923.
[56] A., Ferber, R., Nenadov, and U., Peter, Universality of random graphs and rainbow embedding, Random Structures Algorithms 48 (2016), no. 3, 546–564.
[57] W. Fernandez de la, Vega, Long paths in random graphs, Studia Sci. Math. Hungar. 14 (1979), no. 4, 335–340.
[58] W. Fernandez de la, Vega, Trees in sparse random graphs, J. Combin. Theory Ser. B 45 (1988), no. 1, 77–85.
[59] P., Frankl and V., Rödl, Large triangle-free subgraphs in graphs without K4, Graphs Combin. 2 (1986), no. 2, 135–144.
[60] E., Friedgut, Sharp thresholds of graph properties, and the k-sat problem, J. Amer. Math. Soc. 12 (1999), no. 4, 1017–1054, With an appendix by Jean Bourgain.
[61] E., Friedgut, Hunting for sharp thresholds, Random Structures Algorithms 26 (2005), no. 1-2, 37–51.
[62] J., Friedman and N., Pippenger, Expanding graphs contain all small trees, Combinatorica 7 (1987), no. 1, 71–76.
[63] A., Frieze and M., Karoński, Introduction to random graphs, Cambridge University Press, 2015.
[64] A., Frieze and M., Krivelevich, On two Hamilton cycle problems in random graphs, Israel J. Math. 166 (2008), 221–234.
[65] Z., Füredi, Random Ramsey graphs for the four-cycle, Discrete Math. 126 (1994), no. 1-3, 407–410.
[66] S., Gerke, Random graphs with constraints, 2005, Habilitationsschrift, Institut für Informatik, TU München.
[67] S., Gerke, Y., Kohayakawa, V., Rödl, and A., Steger, Small subsets inherit sparse ∊-regularity, J. Combin. Theory Ser. B 97 (2007), no. 1, 34–56.
[68] S., Gerke and A., McDowell, Nonvertex-balanced factors in random graphs, J. Graph Theory 78 (2015), no. 4, 269–286, (arXiv:1304.3000).
[69] S., Gerke, H. J., Prömel, T., Schickinger, A., Steger, and A., Taraz, K4- free subgraphs of random graphs revisited, Combinatorica 27 (2007), no. 3, 329–365.
[70] S., Gerke, T., Schickinger, and A., Steger, K5-free subgraphs of random graphs, Random Structures Algorithms 24 (2004), no. 2, 194–232.
[71] Y., Gurevich and S., Shelah, Expected computation time for Hamiltonian path problem, SIAM J. Comput. 16 (1987), no. 3, 486–502.
[72] A., Hajnal and E., Szemerédi, Proof of a conjecture of P. Erdʺos, Combinatorial theory and its applications, II (Proc. Colloq., Balatonf üred, 1969), North-Holland, Amsterdam, 1970, pp. 601–623.
[73] P. E., Haxell, Tree embeddings, J. Graph Theory 36 (2001), no. 3, 121–130.
[74] P. E., Haxell, Y., Kohayakawa, and T., Łuczak, Turán's extremal problem in random graphs: forbidding even cycles, J. Combin. Theory Ser. B 64 (1995), no. 2, 273–287.
[75] P. E., Haxell, Y., Kohayakawa, and T., Łuczak, Turán's extremal problem in random graphs: forbidding odd cycles, Combinatorica 16 (1996), no. 1, 107–122.
[76] D., Hefetz, M., Krivelevich, and T., Szabó, Hamilton cycles in highly connected and expanding graphs, Combinatorica 29 (2009), no. 5, 547–568.
[77] D., Hefetz, M., Krivelevich, and T., Szabó, Sharp threshold for the appearance of certain spanning trees in random graphs, Random Structures Algorithms 41 (2012), no. 4, 391–412.
[78] H., Huang, C., Lee, and B., Sudakov, Bandwidth theorem for random graphs, J. Combin. Theory Ser. B 102 (2012), no. 1, 14–37.
[79] S., Janson, T., Łuczak, and A., Ruciński, Random graphs, Wiley- Interscience, New York, 2000.
[80] D., Johannsen, M., Krivelevich, and W., Samotij, Expanders are universal for the class of all spanning trees, Combin. Probab. Comput. 22 (2013), no. 2, 253–281.
[81] A., Johansson, J., Kahn, and V., Vu, Factors in random graphs, Random Structures Algorithms 33 (2008), no. 1, 1–28.
[82] J., Kahn and G., Kalai, Thresholds and expectation thresholds, Combin. Probab. Comput. 16 (2007), no. 3, 495–502.
[83] J., Kahn, E., Lubetzky, and N., Wormald, Cycle factors and renewal theory, Comm. Pure Appl. Math., accepted, arXiv:1401.2707.
[84] J., Kahn, E., Lubetzky, and N., Wormald, The threshold for combs in random graphs, Random Structures Algorithms 48 (2016), no. 4, 794–802.
[85] J. H., Kim and S. J., Lee, Universality of random graphs for graphs of maximum degree two, SIAM J. Discrete Math. 28 (2014), no. 3, 1467–1478.
[86] J. H., Kim and V. H., Vu, Concentration of multivariate polynomials and its applications, Combinatorica 20 (2000), no. 3, 417–434.
[87] J. H., Kim and V. H., Vu, Sandwiching random graphs: universality between random graph models, Adv. Math. 188 (2004), no. 2, 444–469.
[88] Y., Kohayakawa, Szemerédi's regularity lemma for sparse graphs, Foundations of computational mathematics, Springer, 1997, pp. 216– 230.
[89] Y., Kohayakawa, B., Kreuter, and A., Steger, An extremal problem for random graphs and the number of graphs with large even-girth, Combinatorica 18 (1998), no. 1, 101–120.
[90] Y., Kohayakawa, T., Łuczak, and V., Rödl, On K4-free subgraphs of random graphs, Combinatorica 17 (1997), no. 2, 173–213.
[91] Y., Kohayakawa and V., Rödl, Regular pairs in sparse random graphs. I, Random Structures Algorithms 22 (2003), no. 4, 359–434.
[92] Y., Kohayakawa and V., Rödl, Szemerédi's regularity lemma and quasi-randomness, Recent advances in algorithms and combinatorics, Springer, 2003, pp. 289–351.
[93] Y., Kohayakawa, V., Rödl, and M., Schacht, The Turán theorem for random graphs, Combin. Probab. Comput. 13 (2004), no. 1, 61–91.
[94] Y., Kohayakawa, V., Rödl, M., Schacht, and E., Szemerédi, Sparse partition universal graphs for graphs of bounded degree, Adv. Math. 226 (2011), no. 6, 5041–5065.
[95] J., Komlós, The blow-up lemma, Combin. Probab. Comput. 8 (1999), no. 1-2, 161–176, Recent trends in combinatorics (Mátraháza, 1995).
[96] J., Komlós, G. N., Sárközy, and E., Szemerédi, Proof of a packing conjecture of Bollobás, Combin. Probab. Comput. 4 (1995), no. 3, 241–255.
[97] J., Komlós, G. N., Sárközy, and E., Szemerédi, Blow-up lemma, Combinatorica 17 (1997), no. 1, 109–123.
[98] J., Komlós, G. N., Sárközy, and E., Szemerédi, An algorithmic version of the blow-up lemma, Random Structures Algorithms 12 (1998), no. 3, 297–312.
[99] J., Komlós, G. N., Sárközy, and E., Szemerédi, Spanning trees in dense graphs, Combin. Probab. Comput. 10 (2001), no. 5, 397–416.
[100] J., Komlós, A., Shokoufandeh, M., Simonovits, and E., Szemerédi, The regularity lemma and its applications in graph theory, Theoretical aspects of computer science (Tehran, 2000), Lecture Notes in Comput. Sci., vol. 2292, Springer, Berlin, 2002, pp. 84–112.
[101] J., Komlós and M., Simonovits, Szemerédi's regularity lemma and its applications in graph theory, Combinatorics, Paul Erdʺos is eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 295–352.
[102] J., Komlós and E., Szemerédi, Limit distribution for the existence of Hamiltonian cycles in a random graph, Discrete Math. 43 (1983), no. 1, 55–63.
[103] A., Korshunov, Solution of a problem of Erdʺos and Rényi on Hamiltonian cycles in nonoriented graphs., Sov. Math., Dokl. 17 (1976), 760–764.
[104] A., Korshunov, Solution of a problem of P. Erdʺos and A. Rényi on Hamiltonian cycles in undirected graphs, Metody Diskretn. Anal. 31 (1977), 17–56.
[105] M., Krivelevich, Triangle factors in random graphs, Combin. Probab. Comput. 6 (1997), no. 3, 337–347.
[106] M., Krivelevich, Embedding spanning trees in random graphs, SIAM J. Discrete Math. 24 (2010), no. 4, 1495–1500.
[107] M., Krivelevich, C., Lee, and B., Sudakov, Resilient pancyclicity of random and pseudorandom graphs, SIAM J. Discrete Math. 24 (2010), no. 1, 1–16.
[108] D., Kühn and D., Osthus, On Pósa's conjecture for random graphs, SIAM J. Discrete Math. 26 (2012), no. 3, 1440–1457.
[109] C., Lee and W., Samotij, Pancyclic subgraphs of random graphs, J. Graph Theory 71 (2012), no. 2, 142–158.
[110] C., Lee and B., Sudakov, Dirac's theorem for random graphs, Random Structures Algorithms 41 (2012), no. 3, 293–305.
[111] T., Łuczak and A., Ruciński, Tree-matchings in graph processes, SIAM J. Discrete Math. 4 (1991), no. 1, 107–120.
[112] C., McDiarmid and B., Reed, On the maximum degree of a random planar graph, Combin. Probab. Comput. 17 (2008), no. 4, 591–601.
[113] R., Montgomery, Embedding bounded degree spanning trees in random graphs, arXiv:1405.6559.
[114] R., Montgomery, Sharp threshold for embedding combs and other spanning trees in random graphs, arXiv:1405.6560.
[115] J. W., Moon, On the maximum degree in a random tree, Michigan Math. J. 15 (1968), 429–432.
[116] R., Nenadov and N., Škorić, Powers of cycles in random graphs and hypergraphs, arXiv:1601.04034.
[117] A., Noever and A., Steger, Local resilience for squares of almost spanning cycles in sparse random graphs, arXiv:1606.02958.
[118] O., Parczyk and Y., Person, Spanning structures and universality in sparse hypergraphs, Random Structures Algorithms, accepted, arXiv:1504.02243.
[119] L., Pósa, Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), no. 4, 359–364.
[120] O., Riordan, Spanning subgraphs of random graphs, Combin. Probab. Comput. 9 (2000), no. 2, 125–148.
[121] V., Rödl and A., Ruciński, Perfect matchings in ε-regular graphs and the blow-up lemma, Combinatorica 19 (1999), no. 3, 437–452.
[122] V., Rödl, A., Ruciński, and A., Taraz, Hypergraph packing and graph embedding, Combin. Probab. Comput. 8 (1999), no. 4, 363–376, Random graphs and combinatorial structures (Oberwolfach, 1997).
[123] A., Ruciński, Matching and covering the vertices of a random graph by copies of a given graph, Discrete Math. 105 (1992), no. 1-3, 185– 197.
[124] D., Saxton and A., Thomason, Hypergraph containers, Invent. Math. 201 (2015), no. 3, 925–992.
[125] M., Schacht, Extremal results for random discrete structures, Ann. of Math. (2) 184 (2016), no. 2, 333–365.
[126] A., Scott, Szemerédi's regularity lemma for matrices and sparse graphs, Combin. Probab. Comput. 20 (2011), no. 3, 455–466.
[127] E., Shamir, How many random edges make a graph Hamiltonian?, Combinatorica 3 (1983), no. 1, 123–131.
[128] J., Spencer, Threshold functions for extension statements, J. Combin. Theory Ser. A 53 (1990), no. 2, 286–305.
[129] B., Sudakov and V. H., Vu, Local resilience of graphs, Random Structures Algorithms 33 (2008), no. 4, 409–433.
[130] T., Szabó and V. H., Vu, Turán's theorem in sparse random graphs, Random Structures Algorithms 23 (2003), no. 3, 225–234.
[131] A., Thomason, A simple linear expected time algorithm for finding a Hamilton path, Discrete Math. 75 (1989), no. 1-3, 373–379, Graph theory and combinatorics (Cambridge, 1988).
[132] P., Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452.