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.
It is known that every complete graph with an odd number 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 of the vertices into a regular polygon, and covers the complete graph in this subset of vertices with Hamiltonian paths that zigzag across the polygon, with each path rotated from each other path by a multiple of . The paths can then all be completed to Hamiltonian cycles by connecting their ends through the remaining vertex.
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.
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