A stability theorem on fractional covering of triangles by edges

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Abstract

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 graph K4 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 ϵ>0, there exists a K4-free graph Gϵ such that τ(Gϵ)>(2ϵ)ν(Gϵ).

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This research was done at the Institute for Pure and Applied Mathematics at UCLA.