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Let G be the dihedral group Dn and Cay(G, S) is the Cayley graph of G with respect to S, and let CG(x) is the centralizer of an element x in G and x is the orbit of x in G. In this paper, we prove that if G act on G by conjugation, the vertex induced subgraph with vertex set CG(x) of the Cayley graph Cay(G, x) is either Hamiltonian or it contain Hamiltonian decompositions. But if n is prime, it is always Hamiltonian. [ABSTRACT FROM AUTHOR]

The hamiltonian cycles and paths in Cayley graphs naturally arise in computer science in the study of word hyperbolic groups and automatic groups. All Cayley graphs over abelian groups are always hamiltonian. However , for Cayley graphs over non-abelian groups, Chen and Quimpo prove in [1] that Cayley graphs over group of order pq, where p and q primes are Hamiltonian and in [2] that Cayley graphs over hamiltonian groups (i.e., non-abelian groups in which every subgroup is normal) are always hamiltonian. In this paper we investigate the existence of complete hamiltonian cycles and hamiltonian paths in the vertex induced subgraphs of Cayley graphs over non-abelian groups. [ABSTRACT FROM AUTHOR]