Graphs and Combinatorics

, Volume 30, Issue 3, pp 627–632 | Cite as

Packing Triangles in K 4-Free Graphs

Original Paper
First Online:
Received:
Revised:

Abstract

Paul Erdős conjectured that every K 4-free graph of order n and size \({k + \lfloor n^2/4\rfloor}\) contains at least k edge disjoint triangles. In this note, we prove that such a graph contains at least 32k/35 + o(n 2) edge disjoint triangles and prove his conjecture for k ≥  0.077n 2.

Keywords

Erdős’ conjecture Edge-disjoint Packing Triangle 

Mathematics Subject Classification

05C35 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005)Google Scholar
  2. 2.
    Dirac G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 2(3), 69–81 (1952)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Du, X.; Shi L.: Graphs with small independence number minimizing the spectral radius. Discret Math Algorithms Appl (to appear)Google Scholar
  4. 4.
    Erdős P.: On a theorem of Rademacher-Turán. Illinois J. Math. 6, 122–127 (1962)MathSciNetGoogle Scholar
  5. 5.
    Erdős P.: On the number of complete subgraphs contained in certain graphs. Casopis Pest. Mat. 94, 290–296 (1969)MathSciNetGoogle Scholar
  6. 6.
    Erdős, P.: Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and Its Applications, Proceedings Conference, 1969, Acadamic Press, Oxford, pp. 97–109 (1971)Google Scholar
  7. 7.
    Fisher D.: Lower bounds on the number of triangles in a graph. J. Graph Theory 13, 505–512 (1989)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Györi E.: On the number of edge disjoint cliques in graphs of given size. Combinatorica 11(3), 231–243 (1991)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Györi, E.: On the number of edge disjoint triangles in graphs of given size, Combinatorics. In: Proceedings of the 7-th Hungarian Combinatorial Colloquium, Eger, pp. 267–276 (1987)Google Scholar
  10. 10.
    Mantel, W.: Problem 28: Solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. Wiskundige Opgaven 10, 60–61 (1907)Google Scholar
  11. 11.
    Nguyen, H.H.: On the problems of edge disjoint cliques in graphs. Master Thesis, Eőtvős Lornd University, Mathematics Department (2005)Google Scholar
  12. 12.
    Razborov A.A.: Flag algebras. J. Symbolic Logic 72, 1239–1282 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Razborov A.A.: On the minimal density of triangles in graphs. Comb. Probab. Comput. 17, 603–618 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Turán P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436–452 (1941)MathSciNetGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina

Personalised recommendations