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

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 $m$ 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. 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 $m(1/4+c\beta^2)$ pairwise edge-disjoint triangles where $\beta$ 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. It is conjectured that such graphs have an asymptotically optimal triangle packing of size $m(1/3-o(1))$. The result is extended to larger cliques and odd cycles.