Abstract
A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the triangles containing any given edge is one. We prove that every graph graph on n vertices with minimum degree at least 0.9n has a fractional triangle decomposition. This improves a result of Garaschuk that the same conclusion holds for graphs with minimum degree at least 0.956n. Together with a recent result of Barber, Kühn, Lo and Osthus, this implies that for all epsilon > 0, every large enough triangle divisible graph on n vertices with minimum degree at least (0.9 + epsilon)n admits a triangle decomposition.