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A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be an arbitrary prime. Folkman [J. Folkman, Regular line-symmetric graphs, J. Combinatorial Theory, 3 (1967) 215-232.] proved that there are no cubic semisymmetric graphs of order 2p or 2p². In this paper, an extension of his result in the case of cubic graphs of order 44p or 44p² is given. By using group theoretic methods, we prove that there are no connected cubic semisymmetric graphs of order 44p or 44p². [ABSTRACT FROM AUTHOR]

It has been conjectured there is a Hamiltonian cycle in every Cayley graph. Interest in this and other closely related questions has grown in the past few years. There have been many papers on the topic, but it is still an open question whether every connected Cayley graph has a Hamiltonian cycle. In this paper, we survey the results, techniques, and open problems in the field. [ABSTRACT FROM AUTHOR]