Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

Let Γ be a connected simple graph, let V(Γ) and E(Γ) denote the vertex-set and the edge-set of Γ, respectively, and let n=|V(Γ)|. For 1≤i≤n, let e_i be the element of elementary abelian group which has 1 in the ith coordinate, and 0 in all other coordinates. Assume that V(Γ)={e_i∣1≤i≤n}. We define a set Ω by Ω={e_i+e_j∣{e_i,e_j}∈E(Γ)}, and let Cay_Γ denote the Cayley graph over with respect to Ω. It turns out that Cay_Γ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and Cay_Γ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained.