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Hamiltonian decompositions of Cayley graphs on Abelian groups. (English) Zbl 0809.05058
The author considers when the Cayley graph $$\text{cay}(A,S)$$, where $$(A,+)$$ is a group, $$A$$ denotes the vertex set and $$S\subseteq A$$, $$0\not\in S$$, determines the edges of the graph ($$xy$$ is an edge if and only if $$x- y\in S\cup- S$$), has a Hamilton decomposition. It is shown that such a decomposition exists: (1) if $$S= \{s_ 1,\dots, s_ k\}$$ is a generating set of $$A$$ such that $$\text{gcd}(\text{ord}(s_ i), \text{ord}(s_ j))= 1$$ for $$i\neq j$$, or a minimal generating set of $$A$$ with $$k= 3$$ and with either two elements of order 2 or one element of prime order; and (2) if $$A$$ is an Abelian group of odd order and $$S= \{s_ 1,s_ 2,s_ 3\}$$ is a minimal generating set of $$A$$.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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##### References:
 [1] Alspach, B., Research problem 59, Discrete math., 50, 115, (1984) [2] Alspach, B.; Bermond, J.-C.; Sotteau, D., Decomposition into cycles I: Hamilton decompositions, (), 9-18 · Zbl 0713.05047 [3] Aubert, J.; Schneider, B., Decomposition de la somme cartésienne d’un cycle et de l’union de deux cycles hamiltonens en cycles hamiltoniens, Discrete math., 38, 7-16, (1982) · Zbl 0475.05057 [4] Bermond, J.-C.; Favaron, O.; Maheo, M., Hamiltonian decomposition of Cayley graphs of degree 4, J. combin. theory ser. B, 46, 142-153, (1989) · Zbl 0618.05032 [5] G. Chartrand and L. Lesniak, Graphs and Diagraphs, 2nd Edition (Wardsworth and Brooks/Cole, Monterey, CA). · Zbl 1057.05001 [6] Foregger, M.F., Hamiltonian decompositions of products of cycles, Discrete math., 24, 251-260, (1978) · Zbl 0398.05055 [7] J.A. Gallian, Contemporary Abstract Algebra (D.C. Heath, Lexington, MA). [8] Marusic, D., Hamiltonian circuits in Cayley graphs, Discrete math., 46, 49-54, (1983) · Zbl 0515.05042
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