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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)
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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

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