Edge-transitive maps of low genus

Orbanić, Alen; Pellicer, Daniel; Pisanski, Tomaž; Tucker, Thomas W.

doi:10.26493/1855-3974.249.3a6

Cited by 17 publications

(17 citation statements)

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“…It would be interesting to carry over the investigations that were performed for similar operations on maps and their symmetry types, such as those by Orbanić et al [6], del Rio Francos [8,9], and Hubard et al [5] to oriented maps and their oriented symmetry type graphs. Another context in which it would be interesting to investigate similar questions would be in the higher ranks provided by maniplexes and oriented maniplexes (cf., Cunningham et al [24]).…”

Section: Discussionmentioning

confidence: 99%

“…In [20], Širáň, Tucker and Watkins showed that each of the 14 types admits a realization by an oriented map. In [6], Orbanić et al showed that the 14 types can naturally be described by 14 symmetry type graphs, shown in Figure 11.…”

Section: Edge-transitive Oriented Mapsmentioning

confidence: 99%

“…Symmetry type graphs have been used previously in the analysis of maps; see, for instance, [5][6][7][8][9] . We extend these results of maps to oriented maps in the obvious way by introducing a useful tool that we call an oriented symmetry type graph (see Figure 3):…”

Section: Definitionmentioning

confidence: 99%

“…Operations on maps with the property that the new map resides in the same surface as the original map have been the subject of active investigation (see [6,7,[10][11][12][13]). If the underlying surface is orientable, we may choose one orientation that induces an orientation of the corresponding map, making it an oriented map.…”

Section: Operations On Oriented Mapsmentioning

confidence: 99%

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Operations on Oriented Maps

2017

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A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.Keywords: map; oriented map; truncation; dual; medial; snub; flag graph; arc graph MSC: 52C20, 05C10, 51M20 Maps and Oriented MapsA map M on a closed surface is a two-cell embedding of a finite connected graph G = (V, E) (see, e.g., [1] or [2]). Equivalently, every map can be viewed as a set of three fixed-point-free involutions r 0 , r 1 and r 2 acting on a set of flags Ω with the property that r 0 r 2 is also a fixed-point-free involution, in which case we denote the map as M = (Ω, r 0 , r 1 , r 2 ); see, for instance, [3] (p. 415). With this second point of view, each map can be described completely using a three-edge colored cubic graph, called the flag graph, whose vertices are elements of Ω, where ω 1 and ω 2 are connected by an edge colored i in the flag graph if and only if r i (ω 1 ) = ω 2 . Generally, only connected flag graphs are considered. The graph G is called the skeleton or the underlying graph of a map M.In some cases, one may relax the conditions and allow fixed points in some of the involutions r 0 , r 1 and r 2 . Such a structure describes a map in a surface with a boundary and with its flag graph containing semi-edges. We call such a map and respective flag graph degenerate.If the surface in which the map resides is orientable, then M is said to be an orientable map. There is a well-known test involving flag graphs to determine whether a given map is orientable: Proposition 1. A map is orientable if and only if its flag graph is bipartite.In a bipartite flag graph, the flags (i.e., vertices) of the map come in two color classes; the vertices in a single color class of flags are called arcs. Restricting attention to one set of arcs corresponds to choosing an orientation of the orientable map, and we call the restricted graph an oriented map; each orientable map gives rise to two oppositely oriented oriented maps. We note that in a map there are four flags per edge of the skeleton, while in an oriented map there are two arcs per edge. That is,

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Section: Discussionmentioning

confidence: 99%

Section: Edge-transitive Oriented Mapsmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Operations On Oriented Mapsmentioning

confidence: 99%

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Operations on Oriented Maps

2017

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“…Thus, for a reflexible maniplex, the connection group and the symmetry group are isomorphic. For more on symmetry in maps see [15,16].…”

Section: Symmetry In Maniplexesmentioning

confidence: 99%

Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

Wilson

2012

Symmetry

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Arc‐transitive maps with underlying Rose Window graphs

Hubard

Ramos-Rivera

Šparl

2020

Journal of Graph Theory

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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...

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Edge-transitive maps of low genus

Cited by 17 publications

References 17 publications

Operations on Oriented Maps

Operations on Oriented Maps

Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

Arc‐transitive maps with underlying Rose Window graphs

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