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Induced Turán Numbers

Part of: Graph theory

Published online by Cambridge University Press:  02 November 2017

PO-SHEN LOH ,
MICHAEL TAIT ,
CRAIG TIMMONS  and
RODRIGO M. ZHOU
PO-SHEN LOH
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: ploh@cmu.edu, mtait@cmu.edu)
MICHAEL TAIT
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: ploh@cmu.edu, mtait@cmu.edu)
CRAIG TIMMONS
Affiliation:
Department of Mathematics and Statistics, California State University Sacramento, Sacramento, CA 95819, USA (e-mail: craig.timmons@csus.edu)
RODRIGO M. ZHOU
Affiliation:
COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21941-450, Brazil (e-mail: rzhou@cos.ufrj.br)
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Abstract

The classical Kővári–Sós–Turán theorem states that if G is an n-vertex graph with no copy of Ks,t as a subgraph, then the number of edges in G is at most O(n2−1/s). We prove that if one forbids Ks,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in a Kr-free graph with no induced copy of Ks,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C69: Dominating sets, independent sets, cliques
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 274 - 288
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Alon, N. and Shikhelman, C. (2016) Many T copies in H-free graphs. J. Combin. Theory Ser. B 121 146–172.CrossRefGoogle Scholar
[2] Balogh, J., Bollobás, B. and Weinreich, D. (2000) The speed of hereditary properties of graphs. J. Combin. Theory Ser. B 79 131–156.CrossRefGoogle Scholar
[3] Bermond, J., Bond, J., Paoli, M. and Peyrat, C. (1983) Graphs and interconnection networks: Diameter and vulnerability. In Surveys in Combinatorics: Proceedings of the Ninth British Combinatorics Conference, Vol. 82 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 1–30.Google Scholar
[4] Boehnlein, E. and Jiang, T. (2012) Set families with a forbidden induced subposet. Combin. Probab. Comput. 21 496–511.CrossRefGoogle Scholar
[5] Bollobás, B. and Győri, E. (2008) Pentagons vs. triangles. Discrete Math. 308 4332–4336.CrossRefGoogle Scholar
[6] Bollobás, B. and Thomason, A. (1997) Hereditary and monotone properties of graphs. In The Mathematics of Paul Erdős II (Graham, R. L. et al., eds), pp. 70–78.Google Scholar
[7] Chudnovsky, M. (2014) The Erdős–Hajnal conjecture: A survey. J. Graph Theory 75 178–190.Google Scholar
[8] Chung, F. R. K., Gyárfás, A., Tuza, Z. and Trotter, W. T. (1990) The maximum number of edges in 2K 2-free graphs of bounded degree. Discrete Math. 81 129–135.Google Scholar
[9] Chung, M., Jiang, T. and West, D. Induced Turán problems: Largest P m -free graphs with bounded degree. Submitted.Google Scholar
[10] Chung, M. and West, D. (1993) Large P 4-free graphs with bounded degree. J. Graph Theory 17 109–116.CrossRefGoogle Scholar
[11] Chung, M. and West, D. (1996) Large 2P 3-free graphs with bounded degree. Discrete Math. 150 69–79.CrossRefGoogle Scholar
[12] Conlon, D. (2012) On the Ramsey multiplicity of complete graphs. Combinatorica 32 171–186.Google Scholar
[13] Erdős, P. (1962) On the number of complete subgraphs contained in certain graphs. Publ. Math. Inst. Hung. Acad. Sci., VII, Ser. A 3 459–464.Google Scholar
[14] Erdős, P. (1964) On extremal problems of graphs and generalized graphs. Israel J. Math. 2 183–190.Google Scholar
[15] Erdős, P., Frankl, P. and Rödl, V. (1986) The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. 2 113–121.CrossRefGoogle Scholar
[16] Erdős, P. and Hajnal, A. (1989) Ramsey-type theorems. Discrete Appl. Math. 25 37–52.Google Scholar
[17] Erdős, P. and Simonovits, M. (1982) Compactness results in extremal graph theory. Combinatorica 2 275–288.Google Scholar
[18] Erdős, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 1087–1091.Google Scholar
[19] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Comput. Math. 2 463–470.Google Scholar
[20] Fisher, D. and Ryan, J. (1992) Bounds on the number of complete subgraphs. Discrete Math. 103 313–320.CrossRefGoogle Scholar
[21] Fox, J. and Sudakov, B. (2011) Dependent random choice. Random Struct. Alg. 38 68–99.CrossRefGoogle Scholar
[22] Füredi, Z. (1996) New asymptotics for bipartite Turán numbers. J. Combin. Theory Ser. A 75 141–144.Google Scholar
[23] Füredi, Z. and Simonovits, M. (2013) The history of degenerate (bipartite) extremal graph problems. In Erdős Centennial (Lovász, L. et al., eds), Vol. 25 of Bolyai Society Mathematical Studies, Springer, pp. 169–264.Google Scholar
[24] Gyárfás, A., Hubenko, A. and Solymosi, J. (2002) Large cliques in C 4-free graphs. Combinatorica 22 269–274.Google Scholar
[25] Győri, E. and Li, H. (2012) The maximum number of triangles in C 2k + 1-free graphs. Combin. Probab. Comput. 21 187–191.Google Scholar
[26] Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. (2013) On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A 120 722–732.Google Scholar
[27] Li, Y., Rousseau, C. and Zang, W. (2001) Asymptotic upper bounds for Ramsey functions. Graphs Combin. 17 123–128.CrossRefGoogle Scholar
[28] Lu, L. and Milans, K. (2015) Set families with forbidden subposets. J. Combin. Theory Ser. A 136 126–142.CrossRefGoogle Scholar
[29] Parsons, T. D. (1973) The Ramsey numbers r(P m , K n ). Discrete Math. 6 159–162.CrossRefGoogle Scholar
[30] Prömel, H. and Steger, A. (1991) Excluding induced subgraphs I: Quadrilaterals. Random Struct. Alg. 2 55–71.Google Scholar
[31] Prömel, H. and Steger, A. (1993) Excluding induced subgraphs II: Extremal graphs. Discrete Appl. Math. 44 283–294.CrossRefGoogle Scholar
[32] Prömel, H. and Steger, A. (1992) Excluding induced subgraphs III: A general asymptotic. Random Struct. Alg. 3 19–31.Google Scholar
[33] Razborov, A. (2010) On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. 24 946–963.Google Scholar
[34] Sidorenko, A. (1995) What we know and what we do not know about Turán numbers. Graphs Combin. 11 179–199.Google Scholar
[35] Simonovits, M. and Sós, V. T. (2001) Ramsey–Turán theory. Discrete Math. 229 293–340.Google Scholar
[36] Sós, V. T. (1969) On extremal problems in graph theory. In Proceedings of the Calgary International Conference on Combinatorial Structures and their Application, Gordon and Breach, NY, pp. 407–410.Google Scholar
[37] Turán, P. (1941) On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok 48 436–452.Google Scholar