Abstract
The component structure of the most general random hypergraphs, with edges of differen sizes, is analyzed. We show that, as this is the case for random graphs, there is a “double jump” in the probable and almost sure size of the greatest component of hypergraphs, when the average vertex degree passes the value 1.
Similar content being viewed by others
References
M. Ajtai, J. Komlós andE. Szemerédi, The longest path in a random graph,Combinatorica,1 (1981), 1–12.
C. Berge,Graphes et Hypergraphes, Dunod, Paris, (1970).
C. Berge,Introduction à la Theorie des Hypergraphes, Le presse de l’Université de Montréal, Seminaire de Math. Superieur, été 1971.
B. Bollobás andP. Erdős, Cliques in random graphs,Math. Proc. Camb. Phil. Soc.,80 (1976), 419–427.
P. Erdős andA. Rényi, On the evolution of random graphs,Publ. of the Math. Inst. of the Hung. Acad. Sci.,5 (1960), 17–61.
P. Erdős,The Art of Counting, Selected Writings, MIT Press, Cambridge/Massachusetts and London/England, 1973.
P. Erdős andJ. Spencer,Probabilistic Methods in Combinatorics. Academic Press, New York, 1974.
W. Fernandez de la Vega, Sur la cardinalité maximum des couplages d’hypergraphes aléatoire uniformes,Discrete Math. 40, (1982), 315–318.
J. Schmidt andE. Shamir, A threshold for perfect matchings ind-pure random hypergraphs,Discrete Math.,45 (1983), 287–295.
J. Schmidt-Pruzan, E. Shamir, andE. Upfal, Random hypergraph coloring algorithms and the weak chromatic number,Journal of Graph Theory (to appear), (Preliminary version: Technical Report (CS83-09) Dept. Appl. Math. Weizmann Inst. of Sc. Rehovot Israel).
J. Schmidt-Pruzan, Probabilistic analysis of strong hypergraph coloring algorithms and the strong chromatic number.Submitted to Discrete Math., (Preliminary version: Technical Report (CS83-10) Dept. Appl. Math. Weizmann Inst. of Sc. Rehovot Israel).
I. Tomescu, Asymptotical estimations for the number of cliques of uniform hypergraphs,Annals of Discrete Math.,11,Studies on graphs and Discrete Programming (1981), (Edited by P. Hansen), North-Holland Publishing Comp., 345–358.