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

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 ≥ 1 100 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.