# Hamiltonian decomposition

Walecki's Hamiltonian decomposition of the complete graph ${\displaystyle K_{9}}$

In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs; in the undirected case, a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected. For such a decomposition to exist in an undirected graph, it must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even.[1]

It is known that every complete graph with an odd number ${\displaystyle n}$ of vertices has a Hamiltonian decomposition. This result, which is a special case of the Oberwolfach problem of decomposing complete graphs into isomorphic 2-factors, was attributed to Walecki by Édouard Lucas in 1892. Walecki's construction places ${\displaystyle n-1}$ of the vertices into a regular polygon, and covers the complete graph in this subset of vertices with ${\displaystyle (n-1)/2}$ Hamiltonian paths that zigzag across the polygon, with each path rotated from each other path by a multiple of ${\displaystyle \pi /(n-1)}$. The paths can then all be completed to Hamiltonian cycles by connecting their ends through the remaining vertex.[2]

The directed case of complete graphs are tournaments. Answering a 1968 conjecture by Kelly, Daniela Kühn and Deryk Osthus proved in 2012 that every sufficiently large regular tournament has a Hamiltonian decomposition.[3]

Testing whether an arbitrary graph has a Hamiltonian decomposition is NP-complete, both in the directed and undirected cases.[4]

## References

1. ^ Bermond, J.-C. (1978), "Hamiltonian decompositions of graphs, directed graphs and hypergraphs", Annals of Discrete Mathematics, 3: 21–28, doi:10.1016/S0167-5060(08)70494-1, MR 0505807
2. ^ Alspach, Brian (2008), "The wonderful Walecki construction", Bulletin of the Institute of Combinatorics and its Applications, 52: 7–20, MR 2394738
3. ^ Kühn, Daniela; Osthus, Deryk (2013), "Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments", Advances in Mathematics, 237: 62–146, arXiv:1202.6219, doi:10.1016/j.aim.2013.01.005, MR 3028574
4. ^ Péroche, B. (1984), "NP-completeness of some problems of partitioning and covering in graphs", Discrete Applied Mathematics, 8 (2): 195–208, doi:10.1016/0166-218X(84)90101-X, MR 0743024