Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-30T20:32:01.135Z Has data issue: false hasContentIssue false
  • English
  • Français

Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

Published online by Cambridge University Press:  27 September 2012

RAPHAEL YUSTER
RAPHAEL YUSTER*
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel (e-mail: raphy@math.haifa.ac.il)
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.

We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).

We extend our result from triangles to larger cliques and odd cycles.

Keywords

Primary 05C70
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 6 , November 2012 , pp. 952 - 962
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]Alon, N. (1996) Bipartite subgraphs. Combinatorica 16 301–311.CrossRefGoogle Scholar
[2]Haxell, P. (1999) Packing and covering triangles in graphs. Discrete Math. 195 251–254.Google Scholar
[3]Haxell, P. E. and Rödl, V. (2001) Integer and fractional packings in dense graphs. Combinatorica 21 13–38.Google Scholar
[4]Kortsarz, G., Langberg, M. and Nutov, Z. (2008) Approximating maximum subgraphs without short cycles. In Proc. 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization, Vol. 5171 of Lecture Notes in Computer Science, Springer, pp. 118–131.Google Scholar
[5]Krivelevich, M. (1995) On a conjecture of Tuza about packing and covering of triangles. Discrete Math. 142 281–286.Google Scholar
[6]Tuza, Z. (1984) Conjecture. In Finite and Infinite Sets: Eger, Hungary, 1981, Vol. 37 of Proc. Colloq. Math. Soc. Janos Bolyai, North-Holland, p. 888.Google Scholar
[7]Yuster, R. (2005) Integer and fractional packing of families of graphs. Random Struct. Alg. 26 110–118.Google Scholar