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On the order and the number of cliques in a random graph. (English) Zbl 0937.05067
Summary: A clique of a graph $$G$$ is a complete subgraph of $$G$$ maximal under inclusion. We study the numbers of cliques of various orders in a random graph $$G_{n,p}$$ and prove that almost all cliques of $$G_{n,p}$$ have the same order, which is approximately $$\log n -\log \log \log n$$, where log means the logarithm to the base $$1/p$$, and estimate the total number of cliques in $$G_{n,p}$$.

##### MSC:
 05C80 Random graphs (graph-theoretic aspects) 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
random graph; order of cliques; number of cliques
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##### References:
 [1] BOLLOBÁS B.: Random Graphs. Academic Press, New York, 1985. · Zbl 0592.05052 [2] BOLLOBÁS B.-ERDŐS P.: Cliques in random graphs. Math. Proc. Cambridge Philos. Soc. 80 (1976), 419-427. · Zbl 0344.05155 · doi:10.1017/S0305004100053056 [3] FARBER M.-HUJTER M., TUZA, ZS.: An upper bound on the number of cliques in a graph. Networks 23 (1993), 207-210. · Zbl 0777.05070 · doi:10.1002/net.3230230308 [4] FÜREDI Z.: The number of maximal independent sets in connected graphs. J. Graph Theory 11 (1987), 463-470. · Zbl 0647.05032 · doi:10.1002/jgt.3190110403 [5] HEDMAN B.: The maximum number of cliques in dense graphs. Discrete Math. 54 (1985), 161-166. · Zbl 0569.05029 · doi:10.1016/0012-365X(85)90077-9 [6] KALBFLEISCH J. G.: Complete subgraphs of random hypergraphs and bipartite graphs. Proc. of 3rd Southeastern Conference on Combinatorics, Graph Theory and Computing, Florida Atlantic University, 1972, pp. 297-304. · Zbl 0272.05126 [7] KORSHUNOV A. D.: The basic properties of random graphs with large numbers of vertices and edges. Uspekhi Mat. Nauk 40 (1985), 107-173. · Zbl 0574.60015 [8] MATULA D. W.: On the complete subgraphs of a random graph. Proc. 2nd Chapel Hill Conf. Combinatorial Math, and its Applications (R. C. Bose et al., Univ. North Carolina, Chapel Hill, 1970, pp. 356-369. · Zbl 0209.28101 [9] MATULA D. W.: The employee party problem. Notices Amer. Math. Soc. 19 (1972), A-382. [10] MATULA D. W.: The largest clique size in a random graph. Technical report CS 7608, Dept. of Computer Science, Southern Methodist University, Dallas, 1976. [11] MOON J. W.-MOSER L.: On cliques in graphs. Israel J. Math. 3 (1965), 22-28. · Zbl 0144.23205 · doi:10.1007/BF02760024 [12] PALMER E. M.: Graphical Evolution: An Introduction to the Theory of Random Graphs. John Wiley, New York, 1985. · Zbl 0566.05002
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