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How many unique vertex transitive polyhedra exist where each vertex has 3 incident edges for polyhedra with n (= # faces) > 3 ?

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The complete list of vertex transitive planar graphs was determined in 1979 by Fleischner and Imrich. See here. To quote:

Theorem 3. The connected, simple, planar vertex-transitive graphs are the single vertex, the single edge, simple circuits and the nets of the uniform convex polyhedra, namely the nets of regular prisms and antiprisms, the Platonic and the Archimedian bodies.

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See

MR0130841 (24 #A695) Reviewed Tutte, W. T. A census of planar triangulations. Canad. J. Math. 14 1962 21–38. 05.65 (52.45)

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    $\begingroup$ You missed "vertex transitive" in the question, I believe. $\endgroup$ Apr 28, 2014 at 2:54
  • $\begingroup$ @BrendanMcKay Yes, I did :( $\endgroup$
    – Igor Rivin
    Apr 28, 2014 at 3:00

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