The Hamilton spaces of Cayley graphs on abelian groups (Q912863)

The subspace of the cycle space \({\mathcal Z}(X)={\mathcal Z}\) of a graph X that is generated by the Hamilton cycles of X is called the Hamilton space of X and is denoted by \({\mathcal H}(X)={\mathcal H}\). The authors investigate relationships between \({\mathcal Z}(X)\) and \({\mathcal H}(X)\) for graphs X that are Cayley graphs of Abelian groups, since these graphs have an abundance of Hamilton cycles. In particular they show that if X is a connected Cayley graph on a finite Abelian group G, then (i) \({\mathcal H}={\mathcal Z}\) when X is either bipartite or has odd order; or (ii) \({\mathcal H}\) has codimension 2 in \({\mathcal Z}\) when X is a prism over a cycle of odd length; or (iii) \({\mathcal H}\) has codimension 1 in \({\mathcal Z}\) in all other situations.