A proof of Ringel’s conjecture

A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs $$2n+1$$ 2 n + 1 times into the complete graph $$K_{2n+1}$$ K 2 n + 1 . In this paper, we prove this conjecture for large n.