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Hamiltonian decompositions of Cayley graphs on abelian groups of odd order. (English) Zbl 0840.05069
The reviewer has posed the problem [Research Problem 59, Discrete Math. 50, 115 (1984)] of whether or not every connected Cayley graph on a finite abelian group has a Hamiltonian decomposition, that is, a partition of the edge set into Hamilton cycles. The Cayley graph $$X(G; S)$$ on the finite abelian group $$G$$ with symbol $$S$$ is the graph whose vertices are labelled by the elements of $$G$$ with an edge joining $$x$$ and $$y$$ if and only if either $$x- y$$ or $$y- x$$ belongs to $$S$$. With this definition, it is assumed that $$s\in S$$ and $$\text{ord}(s)\neq 2$$ imply $$s^{-1}\not\in S$$. The author proves that a Cayley graph $$X(G; S)$$ on an odd order abelian group $$G$$ has a Hamilton decomposition whenever $$S$$ is a minimal generating set of $$G$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C45 Eulerian and Hamiltonian graphs
##### Keywords:
Cayley graph; abelian group; Hamiltonian decomposition
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