Hostname: page-component-76c49bb84f-kjkt6 Total loading time: 0 Render date: 2025-07-03T06:20:26.500Z Has data issue: false hasContentIssue false
  • English
  • Français

Common pairs of graphs

Part of: Graph theory

Published online by Cambridge University Press:  30 June 2025

Natalie Behague Open the ORCID record for Natalie Behague [Opens in a new window] ,
Natasha Morrison  and
Jonathan A. Noel Open the ORCID record for Jonathan A. Noel [Opens in a new window]
Natalie Behague
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada Mathematics Institute, University of Warwick, Coventry, UK
Natasha Morrison
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
Jonathan A. Noel*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
*
Corresponding author: Jonathan A. Noel; noelj@uvic.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimised by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.

Keywords

Extremal graph theorygraph limitsRamsey multiplicityquasirandomnessSidorenko’s Conjectureflag algebrasRamsey theorycommon graphs

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C55: Generalized Ramsey theory

Information

Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 22
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Behague, N., Morrison, N. and Noel, J. A. (2024) Off-diagonal commonality of graphs via entropy. SIAM J. Discrete Math 38(3) 2335–2360.10.1137/23M1625342CrossRefGoogle Scholar
Burr, S. A. and Rosta, V. (1980) On the Ramsey multiplicities of graphs—problems and recent results. J. Graph Theory 4(4) 347–361.10.1002/jgt.3190040403CrossRefGoogle Scholar
Conlon, D. (2012) On the Ramsey multiplicity of complete graphs. Combinatorica 32(2) 171–186.10.1007/s00493-012-2465-xCrossRefGoogle Scholar
Conlon, D., Fox, J. and Sudakov, B. (2010) An approximate version of Sidorenko’s conjecture. Geom. Funct. Anal 20(6) 1354–1366.10.1007/s00039-010-0097-0CrossRefGoogle Scholar
Conlon, D., Fox, J. and Sudakov, B. (2015) Recent developments in graph Ramsey theory, In Surveys in combinatorics, Vol 424 of London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, 49–118, 2015Google Scholar
Conlon, D., Kim, J. H., Lee, C. and Lee, J. (2018) Some advances on Sidorenko’s conjecture. J. Lond. Math. Soc 98(3) 593–608.10.1112/jlms.12142CrossRefGoogle Scholar
Conlon, D. and Lee, J. (2017) Finite reflection groups and graph norms. Adv. Math 315 130–165.10.1016/j.aim.2017.05.009CrossRefGoogle Scholar
Conlon, D. and Lee, J. (2021) Sidorenko’s conjecture for blow-ups. Discrete Anal., 2, 13. https://doi.org/10.19086/da.21472 Google Scholar
Csóka, E., Hubai, T. and Lovász, L. (2023) Locally common graphs. J. Graph Theory 102(3) 472–483.10.1002/jgt.22881CrossRefGoogle ScholarPubMed
Cummings, J. and Young, M. (2011) Graphs containing triangles are not 3-common. J. Comb 2(1) 1–14.Google Scholar
Erdős, P. (1962) On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 7 459–464.Google Scholar
Even-Zohar, C. and Linial, N. (2015) A note on the inducibility of 4-vertex graphs. Graphs Combin 31(5) 1367–1380.10.1007/s00373-014-1475-4CrossRefGoogle Scholar
Fox, J. (2008) There exist graphs with super-exponential Ramsey multiplicity constant. J. Graph Theory 57(2) 89–98.10.1002/jgt.20256CrossRefGoogle Scholar
Fox, J., Pham, H. T. and Zhao, Y. (2021) Common and Sidorenko linear equations. Q. J. Math 72(4) 1223–1234.10.1093/qmath/haaa068CrossRefGoogle Scholar
Fox, J. and Widgerson, Y. (2023) Ramsey multiplicity and the Turán coloring. Adv. Comb. Paper No. 2, 39.Google Scholar
Goodman, A. W. (1959) On sets of acquaintances and strangers at any party. Amer. Math. Monthly 66(9) 778–783.10.1080/00029890.1959.11989408CrossRefGoogle Scholar
Grzesik, A., Lee, J., Lidický, B. and Volec, J. (2022) On tripartite common graphs. Combin. Probab. Comput 31(5) 907–923.10.1017/S0963548322000074CrossRefGoogle Scholar
Hancock, R., Kráľ, D., Krnc, M. and Volec, J. (2023) Toward characterizing locally common graphs. Random Structures Algorithms 62(1) 181–218.10.1002/rsa.21099CrossRefGoogle Scholar
Hatami, H. (2010) Graph norms and Sidorenko’s conjecture. Israel J. Math 175(1) 125–150.10.1007/s11856-010-0005-1CrossRefGoogle Scholar
Hatami, H., Hladký, J., Kráľ, D., Norine, S. and Razborov, A. (2012) Non-three-colourable common graphs exist. Combin. Probab. Comput 21(5) 734–742.10.1017/S0963548312000107CrossRefGoogle Scholar
Jagger, C., Šťovíček, P. and Thomason, A. (1996) Multiplicities of subgraphs. Combinatorica 16(1) 123–141.10.1007/BF01300130CrossRefGoogle Scholar
Kamčev, N., Liebenau, A. and Morrison, N. (2024) On uncommon systems of equations. Israel J. Math 264(1) 331–362.10.1007/s11856-024-2649-2CrossRefGoogle Scholar
Kim, J. H., Lee, C. and Lee, J. (2016) Two approaches to Sidorenko’s conjecture. Trans. Amer. Math. Soc 368(7) 5057–5074.10.1090/tran/6487CrossRefGoogle Scholar
Ko, S. and Lee, J. (2023) Common graphs with arbitrary connectivity and chromatic number. J. Combin. Theory Ser. B 162 223–230.10.1016/j.jctb.2023.06.001CrossRefGoogle Scholar
Kráľ, D., Noel, J. A., Norin, S., Volec, J. and Wei, F. (2022) Non-bipartite $k$ -common graphs. Combinatorica 42(1) 87–114.10.1007/s00493-020-4499-9CrossRefGoogle Scholar
Kráľ, D., Volec, J. and Wei, F. (2022) Common graphs with arbitrary chromatic number. E-print arXiv: 2206.05800v1.Google Scholar
Lovász, L. (2011) Subgraph densities in signed graphons and the local Simonovits-Sidorenko conjecture. Electron. J. Combin 18(1) 21.10.37236/614CrossRefGoogle Scholar
Lovász, L. (2012) Large networks and graph limits, Vol 60 of American Mathematical Society Colloquium Publications. American Mathematical Society. American Mathematical Society, Providence, RI.Google Scholar
Mitrinović, D. S. (1970) Analytic inequalities. Die Grundlehren der mathematischen Wissenschaften, Band 165. Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić.Google Scholar
Noel, J. A. (2020) Non-bipartite $k$ -common graphs, Online seminar talk, youtu.be/BY1_3glImS0, September 2020. Extremal and Probabilistic Combinatorics Webinar, Organizers: Hladký, J., Piguet, D., Volec, J. and Yepremyan, L..Google Scholar
Parczyk, O., Pokutta, S., Spiegel, C. and Szabó, T. (2024) New Ramsey multiplicity bounds and search heuristics. Found. Comput. Math. https://doi.org/10.1007/s10208-024-09675-6.CrossRefGoogle Scholar
Ramsey, F. P. (1929) On a Problem of Formal Logic. Proc. London Math. Soc 30(4) 264–286.Google Scholar
Razborov, A. A. (2007) Flag algebras. J. Symbolic Logic 72(4) 1239–1282.10.2178/jsl/1203350785CrossRefGoogle Scholar
Saad, A. and Wolf, J. (2017) Ramsey multiplicity of linear patterns in certain finite abelian groups. Q. J. Math 68(1) 125–140.Google Scholar
Sidorenko, A. (1993) A correlation inequality for bipartite graphs. Graphs Combin 9(2) 201–204.10.1007/BF02988307CrossRefGoogle Scholar
Sidorenko, A. (1996) Randomness friendly graphs. Random Structures Algorithms 8(3) 229–241.10.1002/(SICI)1098-2418(199605)8:3<229::AID-RSA6>3.0.CO;2-#3.0.CO;2-#>CrossRefGoogle Scholar
Sidorenko, A. F. (104, 1989) Cycles in graphs and functional inequalities. Mat. Zametki 46(5) 72–79.Google Scholar
Szegedy, B. (2015) An information theoretic approach to Sidorenko’s conjecture. E-print arXiv: 1406.6738v3.Google Scholar
Thomason, A. (1989) A disproof of a conjecture of Erdős in Ramsey theory. J. London Math. Soc 39(2) 246–255.10.1112/jlms/s2-39.2.246CrossRefGoogle Scholar
Thomason, A. (1997) Graph products and monochromatic multiplicities. Combinatorica 17(1) 125–134.10.1007/BF01196136CrossRefGoogle Scholar
Versteegen, L. (2023) Linear configurations containing 4-term arithmetic progressions are uncommon. J. Combin. Theory Ser. A 200, Paper No. 105792, 38.10.1016/j.jcta.2023.105792CrossRefGoogle Scholar
Versteegen, L. (2023) Common and Sidorenko equations in Abelian groups. J. Comb 14(1) 53–67.Google Scholar