Hostname: page-component-77f85d65b8-pkds5 Total loading time: 0 Render date: 2026-04-17T16:34:22.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Induced Subgraphs With Many Distinct Degrees

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  01 August 2017

BHARGAV NARAYANAN  and
ISTVÁN TOMON
BHARGAV NARAYANAN
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: b.p.narayanan@dpmms.cam.ac.uk, i.tomon@dpmms.cam.ac.uk)
ISTVÁN TOMON
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: b.p.narayanan@dpmms.cam.ac.uk, i.tomon@dpmms.cam.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let hom(G) denote the size of the largest clique or independent set of a graph G. In 2007, Bukh and Sudakov proved that every n-vertex graph G with hom(G) = O(logn) contains an induced subgraph with Ω(n 1/2) distinct degrees, and raised the question of deciding whether an analogous result holds for every n-vertex graph G with hom(G) = O(nϵ ), where ϵ > 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graph G on n vertices contains an induced subgraph with Ω((n/hom(G))1/2) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed k ∈ ℕ, that if an n-vertex graph G contains no induced subgraph with k distinct degrees, then hom(G)⩾n/(k − 1) − o(n); this bound is essentially best possible.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C07: Vertex degrees

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 1 , January 2018 , pp. 110 - 123
Copyright
Copyright © Cambridge University Press 2017 

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

[1] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.Google Scholar
[2] Barak, B., Rao, A., Shaltiel, T. and Wigderson, A. (2012) 2-source dispersers for n o(1) entropy, and Ramsey graphs beating the Frankl–Wilson construction. Ann. of Math. 176 1483–1543.Google Scholar
[3] Bukh, B. and Sudakov, B. (2007) Induced subgraphs of Ramsey graphs with many distinct degrees. J. Combin. Theory Ser. B 97 612–619.Google Scholar
[4] Caro, Y. (1979) New results on the independence number. Technical report, Tel Aviv University.Google Scholar
[5] Chattopadhyay, E. and Zuckerman, D. (2016) Explicit two-source extractors and resilient functions. In STOC 2016: Proc. 48th Annual ACM Symposium on the Theory of Computing, ACM, pp. 670–683.Google Scholar
[6] Cohen, G. (2016) Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs. In STOC 2016: Proc. 48th Annual ACM Symposium on the Theory of Computing, ACM, pp. 278–284.Google Scholar
[7] Erdős, P. (1947) Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 292–294.CrossRefGoogle Scholar
[8] Erdős, P. (1992) Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) 47 231–240.Google Scholar
[9] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2 463–470.Google Scholar
[10] Erdős, P. and Szemerédi, A. (1972) On a Ramsey type theorem. Period. Math. Hungar. 2 295–299.Google Scholar
[11] Grolmusz, V. (2000) Low rank co-diagonal matrices and Ramsey graphs. Electron. J. Combin. 7 R15.CrossRefGoogle Scholar
[12] Prömel, H. J. and Rödl, V. (1999) Non-Ramsey graphs are clogn-universal. J. Combin. Theory Ser. A 88 379–384.Google Scholar
[13] Shelah, S. (1998) Erdős and Rényi conjecture. J. Combin. Theory Ser. A 82 179–185.Google Scholar
[14] Wei, V. K. (1981) A lower bound on the stability number of a simple graph. Technical memorandum TM 81-11217-9, Bell Laboratories.Google Scholar