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 Z 2 n which has 1 in the i th 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 Z 2 n 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.