Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

• Published in: Discrete mathematics Vol. 309; no. 17; pp. 5426 - 5431
• Main Authors: Qin, Yong, Xiao, Wenjun, Miklavič, Štefko
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 06.09.2009
• Subjects: Cayley graph; Connected graph; Hamilton cycle; Hamilton path; Spectrum of a graph; Connected graph; Hamilton path; Hamilton cycle; Cayley graph; Spectrum of a graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2008.11.032

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.