Abstract
The origins of the theory of random graphs are easy to pin down. Undoubtedly one should look at a sequence of eight papers co-authored by two great mathematicians: Paul Erdős and Alfred Rényi, published between 1959 and 1968:
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References
D. Achlioptas and A. Naor, The two possible values of the chromatic number of a random graph, Ann. of Math. 162(2) (2005), no. 3, 1335–1351.
M. Ajtai, J. Komlós and E. Szemerédi, Topological complete subgraphs in random graphs, Studia. Sci. Math. Hungar. 14 (1979), 293–297.
N. Alon and J. Spencer, The Probabilistic Method, 1992, Wiley.
N. Alon and R. Yuster, Threshold functions forH-factors, Combinatorics, Probability and Computing 2 (1993), 137–144.
A.D. Barbour, Poisson convergence and random graphs, Math. Proc. Cambr. Phil. Soc. 92 (1982), 349–359.
A.D. Barbour, S. Janson, M. Karoński and A. Ruciński, Small cliques in random graphs, Random Structures Alg. 1 (1990), 403–434.
A.D. Barbour, M. Karoński and A.Ruciński, A central limit theorem for decomposable random variables with applications to random graphs, J. Comb. Th.-B 47 (1989), 125–145.
P. Billingsley, Probability and Measure, 1979, Wiley.
B. Bollobás, Threshold functions for small subgraphs, Math. Proc. Cambr. Phil. Soc. 90 (1981), 197–206.
B. Bollobás, Vertices of given degree in a random graph, J. Graph Theory 6 (1982), 147–155.
B. Bollobás, Distinguishing vertices of random graphs, Annals Discrete Math. 13 (1982), 33–50.
B. Bollobás, Almost all regular graphs are Hamiltonian, Europ. J. Combinatorics 4 (1983), 97–106.
B. Bollobás, The evolution of random graphs, Trans. Amer. Math. Soc. 286 (1984), 257–274.
B. Bollobás, Random Graphs, Academic Press, London, 1985.
B. Bollobás, The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.
B. Bollobás and P. Erdős, Cliques in random graphs, Math. Proc. Cambr. Phil. Soc. 80 (1976), 419–427.
B. Bollobás and A. Frieze, On matchings and hamiltonian cycles in random graphs, in: Random Graphs ’83, Annals of Discrete Mathematics 28 (1985), 1–5.
B. Bollobás and A. Thomason, Random graphs of small order, Annals of Discrete Math. 28 (1985), 47–98.
B. Bollobás and A. Thomason, Threshold functions, Combinatorica 7 (1987), 35–38.
B. Bollobás and J.C. Wierman, Subgraph counts and containment probabilities of balanced and unbalanced subgraphs in a large random graph, in: Graph Theory and Its Applications: East and West (Proc. 1st China-USA Intern. Graph Theory Conf.), Eds. Capobianco et al., Annals of the New York Academy of Sciences 576 (1989), 63–70.
R. DeMarco and J. Kahn, Tight upper tail bounds for cliques 41 (2012), 469487.
[DFl] A. Dudek and A. Frieze, Loose Hamilton Cycles in Random k-Uniform Hypergraphs Electronic Journal of Combinatorics, 18 (2011) P48.
A. Dudek and A. Frieze, Tight Hamilton Cycles in Random Uniform Hypergraphs, Random Structures Alg., to appear.
A. Dudek, A. Frieze, P.-S. Loh and S. Speiss, Optimal divisibility conditions for loose Hamilton cycles in random hypergraphs, Electronic Journal of Combinatorics 19 (2012), P44.
P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.
P. Erdős and A. Rényi, On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.
P. Erdős and A. Rényi On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61.
P. Erdős and A. Rényi, On the evolution of random graphs, Bull. Inst. Internat. Statist. 38 (1961), 343–347.
P. Erdős and A. Rényi, On the strength of connectedness of a random graph, Acta Math. Acad. Sci. Hungar. 12 (1961), 261–267.
P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14 (1963), 295–315.
P. Erdős and A. Rényi, On random matrices, Publ. Math. Inst. Hung. Acad. Sci. 8 (1964), 455–461.
P. Erdős and A. Rényi, On the existence of a factor of degree one of a connected random graph, Acta Math. Acad. Sci. Hung. 17 (1966), 359–368.
P. Erdős and A. Rényi On random matrices II, Studia Sci. Math. Hung. 3 (1968), 459–464.
A. Frieze Loose Hamilton Cycles in Random 3-Uniform Hypergraphs Electronic Journal of Combinatorics 17 (2010) N28
A. Frieze and S. Janson, Perfect Matchings in Random s-Uniform Hypergraphs Random Structures and Algorithms 7 (1995) 41–57.
E. N. Gilbert, Random graphs, Annals of Mathematical Statistics 30 (1959), 1141–1144.
E. Godehardt and J. Steinebach, On a lemma of P. Erdős and A. Rényi about random graphs, Publ. Math. 28 (1981), 271–273.
G.R. Grimmett and C.J.H. McDiarmid, On colouring random graphs, Math. Proc. Cambr. Phil. Soc. 77 (1975), 313–324.
S. Janson, Poisson approximation for large deviations, Random Structures & Algorithms 1 (1990), 221–229.
S. Janson, D.E. Knuth, T. Łuczak and B. Pittel, The birth of the giant component, Random Structures & Algorithms 4 (1993), 233–358.
S. Janson and J. Kratochvil, Proportional graphs, Random Structures & Algorithms 2 (1991), 209–224.
S. Janson, T. Łuczak and A.Ruciński, An exponential bound for the probability of nonexistence of a specified subgraph of a random graph, in: Proceedings of Random Graphs ’87, Wiley, Chichester, 1990, 73–87.
S. Janson, T. Łuczak and A. Ruciński, Random Graphs Wiley, (2000).
S. Janson, K. Oleszkiewicz and A. Ruciński, Upper tails for subgraph counts in random graphs, Israel J. Math. 141 (2004), 61–92.
S. Janson and J. Spencer, Probabilistic constructions of proportional graphs, Random Structures & Algorithms 3 (1992), 127–137.
A. Johansson, J. Kahn and V. Vu, Factors in random graphs, Random Structures and Algorithms 33 (2008), 1–28.
M. Karoński, On the number of k-trees in a random graph, Prob. Math. Stat., 2 (1982), 197–205.
M. Karoński and A. Ruciński, On the number of strictly balanced subgraphs of a random graph, in: Graph Theory, Łagów 1981, Lecture Notes in Math. 1018, Springer-Verlag, 1983, 79–83.
J. Kärrman, Existence of proportional graphs, J. Graph Theory 17 (1993), 207–220.
J. H. Kim, Perfect matchings in random uniform hypergraphs, Random Struct. Algorithms 23(2) (2003), 111–132
V.F. Kolchin, On the limit behavior of a random graph near the critical point, Theory Probability Its Appl. 31 (1986), 439–451.
M. Krivelevich, Perfect fractional matchings in random hypergraphs, Random Structures and Algorithms 9 (1996), 317334.
M. Krivelevich, Triangle factors in random graphs, Combinatorics, Probability and Computing 6 (1997), 337347.
J. Komlós and E. Szemerédi, Limit distributions for the existence of Hamilton cycles, Discrete Math. 43 (1983), 55–63.
A.D. Korshunov, A solution of a problem of Erdős and Rényi on Hamilton cycles in non-oriented graphs, Metody Diskr. Anal. 31 (1977), 17–56.
T. Łuczak, The automorphism group of random graphs with a given number of edges, Math. Proc. Camb. Phil. Soc. 104 (1988), 441–449.
T. Łuczak, On the chromatic number of sparse random graphs, Combinatorica 10 (1990), 377–385.
T. Łuczak, Component behavior near the critical point of the random graph process, Random Structures & Algorithms 1 (1990), 287–310.
T. Łuczak, On the equivalence of two basic models of random graphs, in: Proceedings of Random Graphs ’87, Wiley, Chichester, 1990, 151–157.
T. Łuczak, Size and connectivity of the k-core of a random graph, Discrete Math. 91 (1991) 61–68.
T. Łuczak, A note on the sharp concentration of the chromatic number of a random graph, Combinatorica 11 (1991), 295–297.
T. Łuczak, The phase transition in a random graph, Combinatorics, Paul Erdős is Eighty, vol.2 (Dezsä Miklós, Vera T.Sós, Tamás Szõnyi, eds.), Budapest, 1996, Bolyai Society Mathematical Studies 2, 399–422.
T. Łuczak and J.C. Wierman, The chromatic number of random graphs at the double-jump threshold, Combinatorica 9 (1989), 39–49.
T. Łuczak, B. Pittel and J.C. Wierman, The structure of a random graph at the double-jump threshold, Trans. Am. Math. Soc. 341 (1994), 721–728.
T. Łuczak and A. Ruciński, Tree-matchings in random graph processes, SIAM J. Discr. Math 4 (1991), 107–120.
D. W. Matula, The employee party problem, Notices Amer. Math. Soc. 19 (1972), A-382
D. W. Matula, The largest clique size in a random graph, Tech. Rep. Dept. Comput. Sci., Southern Methodist Univ., Dallas, 1976.
D. W. Matula and L. Kucera, An expose-and-merge algorithm and the chromatic number of a random graph, in: Random Graphs ’87 (J. Jaworski, M. Karoński and A. Ruciński, eds.), John Wiley & Sons, New York, 1990, 175–188.
L. Pósa, Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), 359–364.
A. Ruciński, When are small subgraphs of a random graph normally distributed?, Prob. Th. Rel. Fields 78 (1988), 1–10.
A. Ruciński, Small subgraphs of random graphs: a survey, in: Proceedings of Random Graphs ’87, Wiley, Chichester, 1990, 283–303.
A. Ruciński, Matching and covering the vertices of a random graph by copies of a given graph, Discrete Math. 105 (1992), 185–197.
A. Ruciński and A. Vince, Balanced graphs and the problem of subgraphs of random graphs, Congres. Numerantium 49 (1985), 181–190.
K. Schürger, Limit theorems for complete subgraphs of random graphs, Period. Math. Hungar. 10 (1979), 47–53.
J. Schmidt and E. Shamir, A threshold for perfect matchings in random d-pure hypergraphs, Discrete Math. 45 (1983), 287–295.
E. Shamir and J. Spencer, Sharp concentration of the chromatic number of random graphsG n, p , Combinatorica 7 (1987), 121–129.
E. Shamir and E. Upfal, On factors in random graphs, Israel J. Math.39 (1981), 296–302.
E. M. Wright, Asymmetric and symmetric graphs, Glasgow Math. J. 15 (1974), 69–73.
Acknowledgements
We would like to thank Tomasz Łuczak for his invaluable help in updating this paper. We are also grateful to Steve Butler for turning our obsolete amstex file from 1995 into a modern latex file.
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Karoński, M., Ruciński, A. (2013). The Origins of the Theory of Random Graphs. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_23
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