Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T16:40:55.934Z Has data issue: false hasContentIssue false
  • English
  • Français

The Minimum Number of Triangular Edges and a Symmetrization Method for Multiple Graphs

Published online by Cambridge University Press:  23 January 2017

ZOLTÁN FÜREDI  and
ZEINAB MALEKI
ZOLTÁN FÜREDI
Affiliation:
Alfréd Rényi Institute of Mathematics, 13–15 Reáltanoda Street, 1053 Budapest, Hungary (e-mail: z-furedi@illinois.edu)
ZEINAB MALEKI
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (e-mail: zmaleki@math.iut.ac.ir)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an asymptotic formula for the minimum number of edges contained in triangles among graphs with n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied to several graphs simultaneously.

Keywords

Primary 05C35Secondary 05C2205D99
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 525 - 535
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.)

References

[1] Chung, F. and Graham, R. (1998) Erdős on Graphs: His Legacy of Unsolved Problems, A. K. Peters.Google Scholar
[2] Erdős, P. (1997) Some recent problems and results in graph theory. Discrete Math. 164 81–85.Google Scholar
[3] Erdős, P., Faudree, R. J. and Rousseau, C. C. (1992) Extremal problems involving vertices and edges on odd cycles. Discrete Math. 101 23–31.Google Scholar
[4] Füredi, Z. and Maleki, Z. (2014) The minimum number of triangular edges and a symmetrization for multiple graphs. arXiv:1411.0771 Google Scholar
[5] Füredi, Z. and Maleki, Z. A proof and a counterexample for a conjecture of Erdős concerning the minimum number of edges in odd cycles. Manuscript.Google Scholar
[6] Gruslys, V. and Letzter, S. (2016) Minimising the number of triangular edges. arXiv:1605.00528 Google Scholar
[7] Grzesik, A., Hu, P. and Volec, J. (2016) Minimum number of edges that occur in odd cycles. Manuscript.Google Scholar
[8] Motzkin, T. S. and Straus, E. G. (1965) Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math. 17 533–540.Google Scholar
[9] Turán, P. (1941) On an extremal problem in graph theory (in Hungarian). Matematikai és Fizikai Lapok 48 436–452.Google Scholar
[10] Zykov, A. A. (1949) On some properties of linear complexes (in Russian). Mat. Sbornik (NS) 24 163–188.Google Scholar