Resolution of the Oberwolfach problem

TL;DR

This paper proves that for all sufficiently large n, the complete graph can be decomposed into specified factors, solving the Oberwolfach problem and the Hamilton-Waterloo problem in these cases.

Contribution

It extends the resolution of the Oberwolfach problem to all large n and generalizes to more complex factor decompositions, including the Hamilton-Waterloo problem.

Findings

Decomposition of $K_{2n+1}$ into specified factors for large n

Resolution of the Hamilton-Waterloo problem for large n

Generalization to broader factorization types

Abstract

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.