For large n, every n-vertex graph of minimum degree at least 0.852n has a fractional decomposition into triangles, and every K3-divisible such graph has an exact triangle decomposition above that threshold.
Dross, Fractional Triangle Decompositions in Graphs with Larg e Minimum Degree, SIAM J
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On the minimum degree required for a triangle decomposition
For large n, every n-vertex graph of minimum degree at least 0.852n has a fractional decomposition into triangles, and every K3-divisible such graph has an exact triangle decomposition above that threshold.