Full-Symmetric Embeddings of Graphs on Closed Surfaces

AbstractEvery 2-cell embedding of a graph G on a closed surface F 2 admits a group action of auto-homeomorphisms of F 2 of order at most 4 E ( G ) . The embedding of G is full-symmetric if G attainsthe upper bound for the order. We shall discuss on such full-symmetric embeddings and classify thoseon the projective plane, the torus and the Klein bottle. 1. Introduction Weinberg [7] has shown an upper bound for the order of the automorphism group Aut( G ) of a 3-connected planar graph G : Aut( G ) ≤ 4 E ( G ) Furthermore, the equality holds if and only if G is isomorphic to one of the platonic graphs, that is,the 1-skeletons of regular polyhedra. Since every 3-connected planar graph is uniquely and faithfullyembeddable on the sphere, Aut( G ) can be realized as the symmetry group of an embedding of G on thesphere and hence Aut( G ) coincides with the order of symmetries of an embedding of G on the sphere.(See [4] for the uniqueness and the faithfulness of embeddings.) Extending this observation, we shalldiscuss those graphs embedded on closed surfaces that attain such an upper bound.Let