Abstract
Graver and Watkins classified edge-transitive maps on closed surfaces into fourteen types. In this note we study these types for maps in orientable and non-orientable surfaces of small genus, including the Euclidean and hyperbolic plane. We revisit both finite and infinite one-ended edge-transitive maps. For the finite ones we give precise description that should enable their enumeration for a given number of edges. Edge-transitive maps on surfaces with small genera are classified in the paper.
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