Observations and answers to questions about edge-transitive maps

Abstract

A map is a 2-cell embedding of a connected graph or multigraph on a closed surface, and a map is called edge-transitive if its automorphism group has a single orbit on edges. There are 14 classes of edge-transitive maps, determined by the effect of the automorphism group. In this paper we make some observations about these classes, and answer three open questions from a 2001 paper by Širán, Tucker and Watkins, by showing that (a) in each of the classes 1, 2^P, 2^Pex, 3, 4^P and 5^P, there exists a self-dual edge-transitive map, (b) there exists an edge-transitive map with simple underlying graph on an orientable surface of genus g for every integer g ≥ 0, and (c) there exists an orientable surface that carries an edge-transitive map of each of the 14 classes, and indeed that these three things still hold when we insist that both the map and its dual have simple underlying graph. We also give the maximum number of automorphisms of an edge-transitive map on an orientable surface of given genus g > 1, and consider some special cases in which the automorphism group (or its subgroup of orientation-preserving automorphisms) is prescribed. For example, we show that a certain soluble group of order 576 is the smallest group that occurs as the automorphism group of some edge-transitive map in each of the 14 classes.