Let denote the maximum number of edge-disjoint triangles in a graph and denote the minimum total weight of a fractional covering of its triangles by edges. Krivelevich proved that for every graph . This is sharp, since for the complete graph we have and . We refine this result by showing that if a graph has , then contains edge-disjoint -subgraphs plus an additional edge-disjoint triangles. Note that just these ’s and triangles witness that . Our proof also yields that for each -free graph . In contrast, we show that for each , there exists a -free graph such that .