Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-11T01:01:25.377Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Minimal Density of Triangles in Graphs

Published online by Cambridge University Press:  01 July 2008

ALEXANDER A. RAZBOROV
ALEXANDER A. RAZBOROV*
Affiliation:
Institute for Advanced Study, Princeton, USA and Steklov Mathematical Institute, Moscow, Russia (e-mail: razborov@ias.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that where is the integer such that .

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 4 , July 2008 , pp. 603 - 618
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bollobás, B. (1975) Relations between sets of complete subgraphs. In Proc. Fifth British Combin. Conference, pp. 79–84.Google Scholar
[2]Bollobás, B. (1978) Extremal Graph Theory, Academic Press.CrossRefGoogle Scholar
[3]Borgs, C., Chayes, J., Lovász, L., Sós, V. T. and Vesztergombi, K. (2006) Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Manuscript, available at http://arxiv.org/abs/math/0702004.Google Scholar
[4]Erdős, P. (1962) On a theorem of Rademacher–Turán. Illinois J. Math. 6 122–127.CrossRefGoogle Scholar
[5]Erdős, P. (1969) On the number of complete subgraphs contained in certain graphs. Casopis Pest. Mat. 94 290–296.CrossRefGoogle Scholar
[6]Erdős, P. and Spencer, J. (1974) Probabilistic Methods in Combinatorics, Academic Press.Google Scholar
[7]Erdős, P. and Simonovits, M. (1983) Supersaturated graphs and hypergraphs. Combinatorica 3 181–192.CrossRefGoogle Scholar
[8]Felix, D. (2005) Asymptotic relations in flag algebras. Manuscript, available at http://math.ucsd.edu/~dfelix/partialorder.pdf.Google Scholar
[9]Fisher, D. (1989) Lower bounds on the number of triangles in a graph. J. Graph Theory 13 505–512.CrossRefGoogle Scholar
[10]Goldwurm, M. and Santini, M. (2000) Clique polynomials have a unique root of smallest modulus. Inform. Process. Lett. 75 127–132.CrossRefGoogle Scholar
[11]Goodman, A. W. (1959) On sets of acquaintances and strangers at any party. Amer. Math. Monthly 66 778–783.CrossRefGoogle Scholar
[12]Khadžiivanov, N. and Nikiforov, V. (1978) The Nordhaus–Stewart–Moon–Moser inequality. Serdica 4 344–350. In Russian.Google Scholar
[13]Lovász, L. and Simonovits, M. (1983) On the number of complete subgraphs of a graph II. In Studies in Pure Mathematics, Birkhäuser, pp. 459–495.CrossRefGoogle Scholar
[14]Mantel, W. (1907) Problem 28: Solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. Wiskundige Opgaven 10 60–61.Google Scholar
[15]Moon, J. W. and Moser, L. (1962) On a problem of Turán. Magyar. Tud. Akad. Mat. Kutató Int. Közl 7 283–286.Google Scholar
[16]Nikiforov, V. (2007) The number of cliques in graphs of given order and size. Manuscript, available at http://arxiv.org/abs/0710.2305v2 (version 2).Google Scholar
[17]Nordhaus, E. A. and Stewart, B. M. (1963) Triangles in an ordinary graph. Canadian J. Math. 15 33–41.CrossRefGoogle Scholar
[18]Podolskii, V. V. (2006) Master's thesis, Moscow State University, Moscow. In Russian.Google Scholar
[19]Razborov, A. (2007) Flag algebras. J. Symbolic Logic 72 1239–1282.CrossRefGoogle Scholar
[20]Turán, P. (1941) Egy gráfelméleti szélsöértékfeladatról. Mat. és Fiz. Lapok 48 436–453.Google Scholar