Abstract
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 times into the complete graph K_{2n+1}. In this paper, we prove this conjecture for large n.
Highlights
The study of decomposition problems for graphs and hypergraphs has a very long history, going back more than two hundred years to the work of Euler on Latin squares
In 1782, Euler asked for which values of n there is a Latin square which can be decomposed into n disjoint transversals, where a transversal is a collection of cells of the Latin square which do not share the same row, column or symbol
We say that a graph G has a decomposition into copies of a graph H if the edges of G can be partitioned into edge-disjoint subgraphs isomorphic to H
Summary
The study of decomposition problems for graphs and hypergraphs has a very long history, going back more than two hundred years to the work of Euler on Latin squares. In [MPS20] we gave a new approach for embedding large trees (with no degree restrictions) into edge-colourings of complete graphs, and used this to prove Conjecture 1.1 asymptotically. To prove Theorem 1.2, instead of working directly with tree decompositions, or studying graceful labellings, we prove for large n that every ND-coloured K2n+1 contains a rainbow copy of every n-edge tree (see Theorem 2.1). The existence of such a cyclic decomposition was separately conjectured by Kotzig [ROS66]. When dealing with trees with very high degree vertices, we use a completely deterministic approach for finding a rainbow copy of the tree This approach heavily relies on features of the ND-colouring and produces something very close to a graceful labelling of the tree. We hope that further development of our techniques can help overcome this “bounded degree barrier” in other problems as well
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