Progress towards Nash-Williams' conjecture on triangle decompositions

Abstract

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on n vertices with minimum degree at least 0.75n admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.We show that for any graph on n vertices with minimum degree at least (7+2114)n⪅0.82733n admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for all ε>0, every sufficiently large triangle divisible graph on n vertices with minimum degree at least (7+2114+ε)n admits a triangle decomposition.