Symmetric directed Cayley graphs called rotator graphs have been proposed recently in the literature. In addition to a lower diameter and average distance compared to the star graph, hypercube, and k-ary n-cubes, these rotator graphs also have a rich cyclic structure. We identify a variety of disjoint cycles in rotator graphs. An efficient algorithm for Hamiltonian cycles in rotator graphs is presented. This algorithm uses a basic sequence of four generators repeatedly with generators of higher order in between, to obtain Hamiltonian cycle in any n-rotator graph. We study the embedding of undirected cycles and directed cycles in rotator graphs. We also prove that the incomplete rotator graph obtained from the rotator graph is Hamiltonian. The embedding of undirected rings in rotator graphs is shown to have a low average dilation.
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