A long‑standing conjecture from 1970 by Nash‑Williams asked when the edges of a dense graph can be partitioned into edge‑disjoint triangles. In plain terms, the question is: if a graph has enough edges near every vertex, can we break all its edges into triangles so that each edge belongs to exactly one triangle? Nash‑Williams conjectured that this is always possible for large graphs provided every vertex has degree at least three quarters of the total number of vertices. In this paper Michelle Delcourt and Luke Postle prove that conjecture in full.

The authors proceed in three main parts. First they prove a fractional version of the conjecture: any graph on n vertices whose minimum degree is at least 3n/4 admits a fractional triangle decomposition. A fractional triangle decomposition is a relaxed, linear‑algebra version of the problem. Instead of selecting whole triangles that partition edges, one assigns non‑negative weights to each triangle so that the total weight on every edge sums to one. The proof of this step uses a novel approach the authors call “discharging in the dual,” together with tools from linear programming duality (Farkas’ Lemma). Roughly, the method keeps track of a network of weights and redistributes them by simple local rules to reach a valid fractional assignment.

Second, they prove a fractional stability result. This says that if a graph has minimum degree close to 3n/4 but does not admit a fractional triangle decomposition, then the graph must be structurally close to a specific extremal example: the join of two regular graphs each on half the vertices. In other words, any counterexample near the threshold looks almost like the known tight construction. The authors use this stability information together with property‑testing style arguments to control low‑weight fractional decompositions and to prepare the graph for the final step.