Dense graphs with a large triangle cover have a large triangle packing
Raphael Yuster

TL;DR
This paper proves that dense graphs which are difficult to make triangle-free contain a large number of edge-disjoint triangles, improving previous bounds and extending results to larger cliques and cycles.
Contribution
It establishes a new lower bound on the size of triangle packings in dense, triangle-hard graphs, advancing understanding of their structure.
Findings
Dense graphs hard to make triangle-free have > m(1/4 + cβ^2) edge-disjoint triangles
Improves previous bound of m(1/4 - o(1)) for such graphs
Extends results to larger cliques and odd cycles
Abstract
It is well known that a graph with edges can be made triangle-free by removing (slightly less than) 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 edges in order to eliminate all triangles. It is proved 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 pairwise edge-disjoint triangles where is the density of the graph and is an absolute constant. This improves upon a previous bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. It is conjectured that such graphs have an asymptotically optimal triangle packing of size . The result is extended to larger cliques and odd cycles.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications · Advanced Graph Theory Research
