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) ≤ 2ν(G) for every graph G. This is sharp, since for the complete graphK4 we have ν(K4) = 1 and τ (K4) = 2. We refine this result by showing that if a graph G has τ(G) ≥ 2ν(G) − x, then G contains ν(G) − ⌊10x⌋ edge-disjoint K4-subgraphs plus an additional ⌊10x⌋ edge-disjoint triangles. Note that just these K4’s and triangles witness that τ(G) ≥ 2ν(G) − ⌊10x⌋. Our proof also yields that τ(G) ≤ 1.8ν(G) for each K4-free graph G. In contrast, we show that for each ǫ > 0, there exists a K4-free graph Gǫ such that τ(Gǫ) > (2− ǫ)ν(Gǫ).