Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-23T09:06:50.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Supersaturation of even linear cycles in linear hypergraphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  23 June 2020

Tao Jiang  and
Liana Yepremyan
Tao Jiang
Affiliation:
1Department of Mathematics, Miami University, Oxford, OH45056, USA
Liana Yepremyan*
Affiliation:
2Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA, and Department of Mathematics, London School of Economics, London WC2A 2AE, UK
*
*Corresponding author. Email: L.Yepremyan@lse.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).

In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 5 , September 2020 , pp. 698 - 721
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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.)

Footnotes

†

Research supported in part by National Science Foundation grant DMS-1400249 and DMS-1855542. Email: jiangt@miamioh.edu

‡

Research supported by Marie Sklodowska Curie Global Fellowship 846304.

References

Balogh, J., Narayanan, B. and Skokan, J. (2019) The number of hypergraphs without linear cycles. J. Combin. Theory Ser. B 134 309–321.CrossRefGoogle Scholar
Brown, W. G., Erdős, P. and Sós, V. (1973) On the existence of triangulated spheres in -graphs and related problems. Period. Math. Hungar. 3 221–228.Google Scholar
Bondy, A. and Simonovits, M. (1974) Cycles of even length in graphs. J. Combin. Theory Ser. B 16 97–105.CrossRefGoogle Scholar
Bukh, B. and Jiang, Z. (2017) A bound on the number of edges in graphs without an even cycle. Combin. Probab. Comput. 26 1–15.CrossRefGoogle Scholar
Collier-Cartaino, C., Graber, N. and Jiang, T. (2018) Linear Turán numbers of linear cycles and cycle-complete Ramsey numbers. Combin. Probab. Comput. 27 358–386.CrossRefGoogle Scholar
Erdős, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar. 1 51–57.Google Scholar
Erdős, P. and Simonovits, M. (1970) Some extremal problems in graph theory. In Combinatorial Theory and its Applications I (Proc. Colloq. Balatonfüred, 1969), North-Holland, pp. 377–390.Google Scholar
Erdős, P. and Simonovits, M. (1983) Supersaturated graphs and hypergraphs. Combinatorica 3 181–192.CrossRefGoogle Scholar
Erdős, P. and Simonovits, M. (1984) Cube-supersaturated graphs and related problems. In Progress in Graph Theory (Waterloo, Ont., 1982), Academic Press, pp. 203–218.Google Scholar
Erdős, P. and Stone, H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 1087–1091.CrossRefGoogle Scholar
Ergemlidze, B., Győri, E. and Methuku, A. (2019) Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs. J. Combin. Theory Ser. A 163 163–181.CrossRefGoogle Scholar
Faudree, R. and Simonovits, M. (1983) On a class of degenerate extremal graph problems. Combinatorica 3 83–93.CrossRefGoogle Scholar
Faudree, R. and Simonovits, M. Cycle-supersaturated graphs. In preparation.Google Scholar
Füredi, Z. and Jiang, T. (2014) Hypergraph Turán numbers of linear cycles. J. Combin. Theory Ser. A 123 252–270.CrossRefGoogle Scholar
Füredi, Z. and Simonovits, M. (2013) The history of the degenerate (bipartite) extremal graph problems. In Erdős Centennial, Vol. 25 of Bolyai Society Mathematical Studies, Springer, pp. 169–264. See also https://arXiv.org/abs/1306.5167arXiv:1306.5167.Google Scholar
Janson, S., Łuczak, T. and Ruciski, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
Keevash, P. (2011) Hypergraph Turán problems. In Surveys in Combinatorics, Cambridge University Press, pp. 83–140.Google Scholar
Keevash, P. (2014) The existence of designs. https://arXiv.org/abs/1401.3665arXiv:1401.3665.Google Scholar
Kostochka, A., Mubayi, D. and Verstraëte, J. (2015) Turán problems and shadows I: Paths and cycles. J. Combin. Theory Ser. A 129 57–79.CrossRefGoogle Scholar
Morris, R. and Saxton, D. (2016) The number of -free graphs. Adv. Math. 298 534–580.CrossRefGoogle Scholar
Pikhurko, O. (2012) A note on the Turán function of even cycles. Proc. Amer. Math. Soc. 140 3687–3692.Google Scholar
Rödl, V. (1985) On a packing and covering problem. Europ. J. Combin. 6 69–78.CrossRefGoogle Scholar
Ruzsa, I. and Szemerédi, E.Triples systems with no six points carrying three triangles. In Combinatorics II (Keszthely 1976), Vol. 18 of Colloquia Mathematica Societatis János Bolyai, pp. 939–945.Google Scholar
Sidorenko, A. (1991) Inequalities for functionals generated by bipartite graphs (in Russian). Diskret. Mat. 3 50–65. English translation: Discrete Math. Appl. 2 (1992) 489–504.Google Scholar
Sidorenko, A. (1993) A correlation inequality for bipartite graphs. Graphs Combin. 9 201–204.CrossRefGoogle Scholar
Simonovits, M. (1982) Extremal graph problems, degenerate extremal problems, and supersaturated graphs. In Progress in Graph Theory (Waterloo 1982), Academic Press, pp. 419–437.Google Scholar
Verstraëte, J. (2000) On arithmetic progressions of cycle lengths in graphs. Combin. Probab. Comput. 9 369–373.CrossRefGoogle Scholar
Wilson, R. (1972) An existence theory for pairwise balanced designs I: Composition theorems and morphisms. J. Combin. Theory Ser. A 13 220–245.CrossRefGoogle Scholar
Wilson, R. (1972) An existence theory for pairwise balanced designs II: The structure of PBD-closed sets and the existence conjectures. J. Combin. Theory Ser. A 13 246–273.CrossRefGoogle Scholar
Wilson, R. (1975) An existence theory for pairwise balanced designs III: Proof of the existence conjectures. J. Combin. Theory Ser. A 18 71–79.CrossRefGoogle Scholar