On the Minimum Degree Required for a Triangle Decomposition

We prove that, for sufficiently large n, every graph of order n with minimum degree at least 0.852n has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross [SIAM J. Discrete Math., 30 (2016), pp. 36-42] to establish a bound of 0.9n. By a result of Barber, Kuhn, Lo, and Osthus [Adv. Math., 288 (2016), pp. 337-385], our result implies that, for each epsilon > 0, every graph of sufficiently large order n with minimum degree at least (0.852 +epsilon) n has a triangle decomposition if and only if it has all even degrees and number of edges a multiple of three.