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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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