A stability theorem on fractional covering of triangles by edges

Let ν ( G ) denote the maximum number of edge-disjoint triangles in a graph G and τ ∗ ( G ) denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that τ ∗ ( G ) ≤ 2 ν ( G ) for every graph G . This is sharp, since for the complete graph K 4 we have ν ( K 4 ) = 1 and τ ∗ ( K 4 ) = 2 . We refine this result by showing that if a graph G has τ ∗ ( G ) ≥ 2 ν ( G ) − x , then G contains ν ( G ) − ⌊ 10 x ⌋ edge-disjoint K 4 -subgraphs plus an additional ⌊ 10 x ⌋ edge-disjoint triangles. Note that just these K 4 ’s and triangles witness that τ ∗ ( G ) ≥ 2 ν ( G ) − ⌊ 10 x ⌋ . Our proof also yields that τ ∗ ( G ) ≤ 1.8 ν ( G ) for each K 4 -free graph G . In contrast, we show that for each ϵ > 0 , there exists a K 4 -free graph G ϵ such that τ ( G ϵ ) > ( 2 − ϵ ) ν ( G ϵ ) .