Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T08:17:30.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Degree Powers in Graphs: The Erdős–Stone Theorem

Published online by Cambridge University Press:  02 February 2012

BÉLA BOLLOBÁS  and
VLADIMIR NIKIFOROV
BÉLA BOLLOBÁS
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (e-mail: b.bollobas@dpmms.cam.ac.uk) Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA (e-mail: vnikifrv@memphis.edu)
VLADIMIR NIKIFOROV
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA (e-mail: vnikifrv@memphis.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let 1 ≤ p ≤ r + 1, with r ≥ 2 an integer, and let G be a graph of order n. Let d(v) denote the degree of a vertex v ∈ V(G). We show that if then G has more than (r + 1)-cliques sharing a common edge. From this we deduce that if then G contains more than cliques of order r + 1.

In turn, this statement is used to strengthen the Erdős–Stone theorem by using ∑v ∈ V(G)dp(v) instead of the number of edges.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 89 - 105
Copyright
Copyright © Cambridge University Press 2012

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. (1998) Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer.Google Scholar
[2]Bollobás, B. and Nikiforov, V. (2004) Degree powers in graphs with forbidden subgraphs. Electron. J. Combin. 11 R42.CrossRefGoogle Scholar
[3]Bollobás, B. and Nikiforov, V. (2008) Joints in graphs. Discrete Math. 308 9–19.CrossRefGoogle Scholar
[4]Caro, Y. and Yuster, R. (2000) A Turán type problem concerning the powers of the degrees of a graph. Electron. J. Combin. 7 R47.CrossRefGoogle Scholar
[5]Caro, Y. and Yuster, R. A Turán type problem concerning the powers of the degrees of a graph (revised). Preprint. arXiv:math/0401398v1Google Scholar
[6]Erdős, P. (1970) On the graph theorem of Turán (in Hungarian). Mat. Lapok 21 249–251.Google Scholar
[7]Nikiforov, V. (2008) Graphs with many r-cliques have large complete r-partite subgraphs. Bull. London Math. Soc. 40 23–25.CrossRefGoogle Scholar
[8]Nikiforov, V. (2009) Degree powers in graphs with forbidden even cycle. Electron. J. Combin. 11 R107.CrossRefGoogle Scholar
[9]Nikiforov, V. (2009) Stability for large forbidden subgraphs. J. Graph Theory 62 362–368.CrossRefGoogle Scholar
[10]Nikiforov, V. and Rousseau, C. C. (2009) Ramsey goodness and beyond. Combinatorica 29 227–262.CrossRefGoogle Scholar
[11]Pikhurko, O. Remarks on a paper of Y. Caro and R. Yuster on Turán problem. Preprint. arXiv:math.CO/0101235v1Google Scholar
[12]Pikhurko, O. and Taraz, A. (2005) Degree sequences of F-free graphs. Electron. J. Combin. 12 R69CrossRefGoogle Scholar