Sudden emergence of a giant k-core in a random graph

Journal of Combinatorial Theory. Series B

Volumn 67, Issue 1, 1996, Pages 111-151

  Pittel, Boris ^a   Spencer, Joel ^b   Wormald, Nicholas ^c  

^a OHIO STATE UNIVERSITY   (United States)

^b NEW YORK UNIVERSITY   (United States)

^c UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030144334     PISSN: 00958956     EISSN: None     Source Type: Journal    
DOI: 10.1006/jctb.1996.0036     Document Type: Article

References (29)

1. 1. E. A. BENDER AND E. A. CANFIELD, The asymptotic number of labelled graphs with given degree sequences, J. Combin. Theory Ser. A 24 (1978), 296-307.
2. 2. B. BOLLOBÁS, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, European J. Combin. 1 (1980), 311-316.
3. 3. B. BOLLOBÁS, The evolution of sparse graphs, in "Graph Theory and Combinatorics: Proc. Cambridge Combinatorial Conf. in honour of Paul Erdös" (B. Bollobás, Ed.), pp. 35-57, Academic Press, New York, 1984.
4. 4. B. BOLLOBÁS, "Random Graphs," Academic Press, New York, 1985.
5. 5. B. BOLLOBÁS AND A. M. FRIEZE, On matchings and Hamiltonian cycles in random graphs, in "Random Graphs' 83," Annals of Discrete Mathematics, Vol. 28 (M. Karoński and A. Ruciński, Eds.), pp. 23-46, North-Holland, Amsterdam/New York, 1985.
6. 6. B. BOLLOBÁS, Martingales, isoperimetric inequalities and random graphs, in "Combinatorics, Eger (Hungary) 1987," Colloquia Societatis Jánois Bolyai, Vol. 52, pp. 113-139, Akademiai Kiadó, Budapest, 1988.
7. 7. B. BOLLOBÁS AND S. RASMUSSEN, First cycles in random directed graph processes, Discrete Math. 75 (1989), 55-68.
8. 8. V. CHVÁTAL, Almost all graphs with 1.44 edges are 3-colourable, Random Structures Algorithms 2 (1991), 11-28.
9. 9. R. DURRETT, "Probability: Theory and Examples," Wadsworth & Brooks, Pacific Grove, CA, 1991.
10. 10. P. ERDÖS AND A. RÉNYI, On the evolution of random graphs, Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 5 (1960), 17-61.
11. 11. W. FELLER, "An Introduction to Probability Theory and Its Applications, II," 2nd ed., Wiley, New York, 1971.
12. 12. P. FLAJOLET, D. E. KNUTH, AND B. PITTEL, The first cycles in an evolving graph, Discrete Math. 75 (1989), 167-215.
13. 13. G. HAVAS, B. S. MAJEWSKI, N. C. WORMALD, AND Z. J. CZECH, Graphs, hypergraphs and hashing, in "19th International Workshop on Graph-Theoretic Concepts in Computer Science (WG' 93), Utrecht, 1993," Lecture Notes in Computer Science, Vol. 790, pp. 153-165, Springer-Verlag, New York/Berlin, 1994.
14. 14. S. JANSON, Poisson convergence and Poisson processes with application to random graphs, Stochast. Process. Appl. 26 (1987), 1-30.
15. 15. S. JANSON, D. E. KNUTH, T. ŁUCZAK, AND B. PITTEL, The birth of the giant component, Random Structures Algorithms (Issue in Honour of Dr. Paul Erdös) 4 (1993), 233-358.
16. 16. R. M. KARP AND M. SIPSER, Maximum matchings in sparse random graphs, in "IEEE Conference on Computing, 1981," pp. 364-375.
17. 17. R. M. KARP, The transitive closure of a random digraph, Random Structures Algorithms 1 (1990), 73-93.
18. 18. T. ŁUCZAK, Private communication, (1989).
19. 19. T. ŁUCZAK, Component behavior near the critical point of the random graph process, Random Structures Algorithms 1 (1990), 287-310.
20. 20. T. ŁUCZAK, Size and connectivity of the k-core of a random graph, Discrete Math. 91 (1991), 61-68.
21. 21. T. ŁUCZAK, Sparse random graphs with a given degree sequence, in "Random graphs, 2" (A. Frieze and T. Łuczak, Eds.), pp. 165-182, Wiley, New York, 1992.
22. 22. T. ŁUCZAK, B. PITTEL, AND J. C. WIERMAN, The structure of a random graph at the point of the phase transition, Trans. Amer. Math. Soc. 341 (1994), 721-748.
23. 1/2), Combinatorica 11 (1991), 369-382.
24. 24. B. S. MAJEWSKI, N. C. WORMALD, G. HAVAS, AND Z. J. CZECH, A family of perfect hashing methods, submitted for publication.
25. 25. M. MOLLOY, M. Math. Thesis, University of Waterloo, 1992.
26. 26. B. PITTEL, On a Daley-Kendall model of random rumours, J. Appl. Probab. 27 (1990), 14-27.
27. 27. B. PITTEL, On trees census and the giant component in sparse random graphs, Random Structures Algorithms 1 (1990), 311-342.
28. 28. B. PITTEL AND W. A. WOYCZYNSKI, A graph-valued Markov process as rings-allowed polymerization model: Subcritical behavior, SIAM J. Appl. Math. 50 (1990), 1200-1220.
29. 29. N. C. WORMALD, Differential equations for random processes and random graphs, submitted for publication.