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On the Maximum Number of Cliques in a Graph

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Abstract

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K5-minor or no K3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2).

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References

  • Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth, Theor. Comput. Sci. 209(1–2), 1–45 (1998)

    Google Scholar 

  • Bollobás, B.: Extremal graph theory. In: Handbook of Combinatorics, vol. 1, 2, pp. 1231–1292, Elsevier, 1995

  • Bollobás, B., Erdős, P.: Cliques in random graphs, Math. Proc. Cambridge Philos. Soc. 80(3), 419–427 (1976)

    Google Scholar 

  • Bollobás, B., Erdős, P., Szemerédi, E.: On complete subgraphs of r-chromatic graphs, Disc. Math. 13(2), 97–107 (1975)

    Google Scholar 

  • Eckhoff, J.: The maximum number of triangles in a K4-free graph, Disc. Math. 194(1–3), 95–106 (1999)

    Google Scholar 

  • Eckhoff, J.: A new Turán-type theorem for cliques in graphs, Disc. Math. 282(1–3), 113–122 (2004)

    Google Scholar 

  • Eppstein, D.: Connectivity, graph minors, and subgraph multiplicity, J. Graph Theory 17(3), 409–416 (1993)

    Google Scholar 

  • Erdős, P.: On the number of complete subgraphs contained in certain graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7, 459–464 (1962)

  • Erdős, P.: On cliques in graphs, Israel J. Math. 4, 233–234 (1966)

    Google Scholar 

  • Erdős, P.: On the number of complete subgraphs and circuits contained in graphs. Časopis Pěst. Mat. 94, 290–296 (1969)

    Google Scholar 

  • Fan, C., Liu, J.: On cliques of graphs. In: Graph Theory, Combinatorics, Algorithms, and Applications, pp. 140–150. SIAM (1991)

  • Farber, M., Hujter, M., Tuza, Z.: An upper bound on the number of cliques in a graph, Networks 23(3), 207–210 (1993)

    Google Scholar 

  • Fisher, D.C.: Lower bounds on the number of triangles in a graph, J. Graph Theory 13(4), 505–512 (1989)

    Google Scholar 

  • Fisher, D.C., Nonis, J.M.: Bounds on the “growth factor” of a graph. In: Proc. 21st Southeastern Conf. on Combinatorics, Graph Theory, and Computing vol. 77 of Congr. Numer., pp. 113–119. Utilitas Math. 1990

  • Fisher, D.C., Ryan, J.: Bounds on the number of complete subgraphs, Disc. Math. 103(3), 313–320 (1992)

    Google Scholar 

  • Hadžiivanov, N.G.: Maximum of a multilinear form of integer variables, and its application in the theory of extremal graphs, C. R. Acad. Bulgare Sci. 30(10), 1373–1376 (1977)

  • Hadžiivanov, N.G., Nenov, N.D.: p-sequences of graphs and some extremal properties of Turán graphs, C. R. Acad. Bulgare Sci. 30(4), 475–478 (1977)

    Google Scholar 

  • Harary, F., Lempel, A.: On clique-extremal (p,q)-graphs, Networks 4, 371–378 (1974)

  • Jou, M.-J., Chang, G.J.: The number of maximum independent sets in graphs, Taiwanese J. Math. 4(4), 685–695 (2000)

    Google Scholar 

  • Khadzhiivanov, N., Nenov, N.: Sharp estimates of the highest number of cliques of a graph, Ann. Univ. Sofia Fac. Math. Méc. 70, 23–26 (1975/76)

    Google Scholar 

  • Kostochka, A.V.: The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret. Analiz. 38, 37–58 (1982)

    Google Scholar 

  • Larman, D.G.: On the number of complete subgraphs and circuits in a graph, Proc. Roy. Soc. Ser. A 308, 327–342 (1969)

    Google Scholar 

  • Lovász, L., Simonovits, M.: On the number of complete subgraphs of a graph. In: Proc. of 5th British Combinatorial Conference, vol. XV of Congr. Numer. pp. 431–441. Utilitas Math. 1976

  • Lovász, L., Simonovits, M.: On the number of complete subgraphs of a graph. II: In: Studies in Pure Mathematics, pp. 459–495. Birkhäuser, 1983

  • Moon, J.W.: On the number of complete subgraphs of a graph, Can. Math. Bull. 8, 831–834 (1965)

    Google Scholar 

  • Moon, J.W., Moser, L.: On cliques in graphs, Israel J. Math. 3, 23–28 (1965)

    Google Scholar 

  • Nenov, N., Khadzhiivanov, N.: A new proof of a theorem about the maximum number of p-cliques of graphs, Ann. Univ. Sofia Fac. Math. Méc. 70, 93–98 (1975/76)

    Google Scholar 

  • Norine, S., Seymour, P., Thomas, R., Wollan, P.: Proper minor-closed families are small, J. Combin. Theory Ser. B 96(5), 754–757 (2006)

    Google Scholar 

  • Olejár, D., Toman, E.: On the order and the number of cliques in a random graph, Math. Slovaca 47(5), 499–510 (1997)

    Google Scholar 

  • Papadimitriou, C.H., Yannakakis, M.: The clique problem for planar graphs, Inform. Process. Lett. 13(4–5), 131–133 (1981)

    Google Scholar 

  • Petingi, L., Rodriguez, J.: A new proof of the Fisher-Ryan bounds for the number of cliques of a graph. In: Proc. 31st Southeastern International Conf. on Combinatorics, Graph Theory and Computing, vol. 146 of Congr. Numer. pp. 143–146, 2000

  • Reed, B., Wood, D.R.: Fast separation in a graph with an excluded minor. In: Proc. European Conf. on Combinatorics, Graph Theory and Applications (EuroComb ’05), vol. AE of Discrete Math. Theor. Comput. Sci. Proceedings, pp. 45–50, 2005

  • Roman, S.: The maximum number of q-cliques in a graph with no p-clique, Disc. Math. 14(4), 365–371 (1976)

    Google Scholar 

  • Sagan, B.E., Vatter, V.R.: Maximal and maximum independent sets in graphs with at most r cycles, J. Graph Theory 53(4), 283–314 (2006)

    Google Scholar 

  • Sato, I.: Clique graphs of packed graphs, Disc. Math. 62(1), 107–109 (1986)

    Google Scholar 

  • Sauer, N.: A generalization of a theorem of Turán, J. Comb. Theory Ser. B 10, 109–112 (1971)

    Google Scholar 

  • Schürger, K.: Limit theorems for complete subgraphs of random graphs, Period. Math. Hungar. 10(1), 47–53 (1979)

    Google Scholar 

  • Simonovits, M.: Extremal graph theory. In: Selected Topics in Graph Theory, vol. 2, pp. 161–200, Academic Press, 1983

  • Simonovits, M.: Paul Erdős’ influence on extremal graph theory. In The Mathematics of Paul Erdős, II vol. 14 of Algorithms Combin., pp. 148–192, Springer, 1997

  • Simonovits, M., Sós, V.T.: Ramsey-Turán theory, Disc. Math. 229(1–3), 293–340 (2001)

    Google Scholar 

  • Sitton, D.: Maximum matchings in complete multipartite graphs, Furman Univ. Electronic J. Undergraduate Math. 2, 6–16 (1996)

    Google Scholar 

  • Spencer, J.H.: On cliques in graphs, Israel J. Math. 9, 419–421 (1971)

    Google Scholar 

  • Stiebitz, M.: On Hadwiger’s number—a problem of the Nordhaus-Gaddum type, Disc. Math. 101(1–3), 307–317 (1992)

    Google Scholar 

  • Storch, T.: How randomized search heuristics find maximum cliques in planar graphs. In: Proc. of 8th Annual Conf. on Genetic and Evolutionary Computation (GECCO ’06), pp. 567–574, ACM Press, 2006

  • Thomason, A.: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95(2), 261–265 (1984)

    Google Scholar 

  • Thomason, A.: The extremal function for complete minors, J. Combin. Theory Ser. B 81(2), 318–338 (2001)

    Google Scholar 

  • Turán, P.: On an extremal problem in graph theory, Mat. Fiz. Lapok 48, 436–452 (1941)

    Google Scholar 

  • Wagner, K.: Über eine Eigenschaft der ebene Komplexe, Math. Ann. 114, 570–590 (1937)

    Google Scholar 

  • West, D.B.: The number of complete subgraphs in graphs with nonmajorizable degree sequences. In: Progress in Graph Theory, pp. 509–521, Academic Press, 1984

  • Wilf, H.S.: The number of maximal independent sets in a tree, SIAM J. Algebraic Disc. Methods 7(1), 125–130 (1986)

    Google Scholar 

  • Ying, G.C., Meng, K.K., Sagan, B.E., Vatter, V.R.: Maximal independent sets in graphs with at most rcycles, J. Graph Theory 53(4), 270–282 (2006)

    Google Scholar 

  • Zykov, A.A.: On some properties of linear complexes, Mat. Sbornik N.S. 24(66), 163–188 (1949)

    Google Scholar 

Download references

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Authors and Affiliations

  1. Departament de Matemática Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    David R. Wood

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  1. David R. Wood
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Correspondence to David R. Wood.

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Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.

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Wood, D.R. On the Maximum Number of Cliques in a Graph. Graphs and Combinatorics 23, 337–352 (2007). https://doi.org/10.1007/s00373-007-0738-8

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