Hamiltonian decompositions of Cayley graphs on Abelian groups (Q1332424)

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 \(A\).