Arc‐transitive maps with underlying Rose Window graphs

Abstract

In the late 1990s, Graver and Watkins initiated the study of all edge-transitive maps. Recently, Gareth Jones revisited the study of such maps and suggested classifying the maps in terms of either their automorphism groups or their underlying graphs. A natural step towards classifying edge-transitive maps is to study the arc-transitive ones. In this paper, we investigate the connection of a class of arc-transitive maps to consistent cycles of the underlying graph, with special emphasis on maps of smallest possible valence, namely 4. We give a complete classification of arc-transitive maps whose underlying graphs are arc-transitive Rose Window graphs. K E Y W O R D S arc-transitive graph, arc-transitive map, consistent cycle, 2-orbit map maps, based on the local configuration of the flags (incident vertex-edge-face triples) and the orbits they belong to (see also [24]). Two of these 14 classes correspond to rotary maps. Even though most of the papers on edge-transitive maps in the existing literature focus on these two classes of rotary maps, some papers also investigated other classes (see, for instance [19, 23]). In these papers questions like which edge-transitive maps of small genera exist, and which of the 14 types of edge-transitive maps can be realized by maps with an automorphism group abstractly isomorphic to a symmetric group, have been considered. Recently, Jones (see [12]) extended the work of [23] while working on the question of which groups can act as the automorphism group of an edge-transitive map by answering it for some specific classes of groups. This question naturally extends to the question of which graphs can be the underlying graph of an edge-transitive map. Moreover, in both cases one can ask which are the classes of edge-transitive maps that can occur, how many maps of a given class are there and whether one can describe all of them. When studying edge-transitive maps it is natural to restrict to a subset of them: those whose automorphism group acts transitively on the set of arcs. Rotary maps are an example of such maps, but again, there are others. For example, the cuboctahedron (or medial of the cube) seen as a map on the sphere is arc-transitive, but it is not a rotary map. As already mentioned, rotary maps can be divided into two different classes: one that contains maps admitting involutory automorphisms that fix a given arc but interchange the two faces incident to it (maps in this class are called regular or reflexible maps), and the other, which contains the rest of the rotary maps (maps in this class are called chiral). The automorphism group of a reflexible map is transitive on the set of its flags, while the automorphism group of a chiral map has two orbits on the set of its flags. As we shall see, it is not difficult to prove that there are exactly five classes of arc-transitive maps. When one applies the Petrie dual operator to a map, the underlying graph, as well as the automorphism group of the map, remain unchanged. Hence, classes of arc-transitive maps are linked via th...

Symmetries of the Woolly Hat graphs

“…It has taken a considerable effort of several researchers in a handful of papers before the last step of the classification was obtained in [13]. For instance, one of the major steps was the classification of edge-transitive Rose window graphs [11], a family of graphs intro-duced in [27] that have since been the object of investigation in various contexts (see for instance [9]).…”