Christofides, Hladky, and Mathe (http://arxiv.org/abs/1008.2193) proved using the Regularity Method the case when the vertex-transitive graph is sufficiently dense.
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Hamiltonicity of dence Cayley graphs
Christofides, Hladky, and Mathe (http://arxiv.org/abs/1008.2193) proved using the Regularity Method the case when the vertex-transitive graph is sufficiently dense.