Edge-transitive maps on orientable surfaces

Here is the list of isomorphism classes of oriented edge-transitive maps on the orientable surfaces of genus \( g > 1\). The list contains all the oriented maps of small genera such that groups of orientation-preserving automorphisms have at most two orbits on the set of edges of the corresponding map. The two orbits are interchanged by an orientation-reversing automorphism of the map. Thanks to this strong conditions one can use the existing lists of discrete group actions (see e.g here) to derive full census of oriented edge-transitive maps of given genus.

The census consists of plain text files, a file for a genus. The format of record is provided with hope that records are self-explanatory. However, here is the description of data format used.

Data format

The first line of the file begins with the tex ## Begin on: followed by the timestamp of the creation of the file. The line before the last line contains the number of the records in the file introduced by the text ## Checksum:. The first line of the file begins with the tex ## Written on: followed by the timestamp when file has been commited to the storage.

The file continue with the section ENUMERATION. The maps are counted with respect to the families defined and used in [GW, OPPT, STW]. Counts of non-degenerate edge-transitive maps are present. The map with simple underlying graph is non-degenerate if every vertex, face and Petrie-walk has degree (length) at least \(3\) [OPPT].

The content of the file consists of the records separated by an empty line and begins immediately after the last line of the enumeration (which counts the maps of family \(5^P\)). The record consists of at most 17 text lines and contains the following data

1. The header line consists of three fields separated with ::. The first field holds the identifier of the map in the form Eg.n, where g is the genus \(g\) of the edge-transitive map \(\mathbf{M}\) and n is the unique index in the file. This tag uniquely determines the isomorphism class with respect to \(\operatorname{Aut}\,\mathbf{M}\) of edge-transitive maps in the census. Note that the map and its mirror image belong to the same isomorphism class. The second field in the line contains the identifier of the family of edge-transitive maps defined in [GW, OPPT, STW]. The last field serves just for debug purposes; it keeps the information about the oriented family of the map as introduced in [KS]. Roughly, the correspondence among the classes (families) used in [GW, OPPT, STW] and in our draft [KS] is given by the following table
   

2. The signature of the corresponding quotient orbifold \(\mathcal{O}(\gamma,\{m_1,m_2,\ldots,m_r\}) = \mathcal{S}_g/\mathrm{Aut}^+\,\mathbf{M}\), see e.g. here.
3. The definition of the quotient of the map \(\mathbf{M}/\mathrm{Aut}^+\,\mathbf{M}\). The quotient map is described by the string as follows from the following table
   

4. The group of orientation-preserving automorphisms, \(\mathrm{Aut}^+\,\mathbf{M}\), of the map. The line contains the abstract (structural) description of the group (as computed by StructureDescription by GAP [GAP]) and the identifier of the group in SmallGroups library [BEB]. Note that that this group has action on the corresponding orientable surface of genus \(g\) and the corresponding signature of the orbifold \(\mathcal{O}\) given in Line 2 following from the Riemmann-Hurwitz equation, \[ 2g-2 = |\mathrm{Aut}^+\,\mathbf{M}|\left[2\gamma-2 + \sum_{i=1}^r \left(1-\frac{1}{m_i}\right)\right], \] where \(\gamma\) is the genus the orbifold and \(\{m_1,m_2,\ldots,m_r\}\) is the multiset of branch indexes.
5. The automorphism group, \(\mathrm{Aut}\,\mathbf{M}\), of the map. Again, the line contains the abstract description of the group and the identifier of the group in SmallGroups library [BEB].
6. The order of orientation reversing automorphism. The element \(r\in\mathrm{Aut}\,\mathbf{M}\) brings the transitivity of \(r\in\mathrm{Aut}\,\mathbf{M}\) on the set of edges of \(\mathbf{M}\). We have \(r\notin\mathrm{Aut}^+\,\mathbf{M} \wedge r^2\in\mathrm{Aut}^+\,\mathbf{M} \) and \(r\) is not necessarily (but it is mostly) an involution. In case of maps of family \(5^P\) the reflection acts mostly as a glide reflection ( \(r\) does not have order \(2\)). The reflection is degenerated to the identity, if the quotient itself has one (semi-)edge; which is the case of maps in families \(1\), \(5\) and \(5^*\).
7. The presentation of the group of automorphisms with transitive action on edges of \(\mathbf{M}\). The presentation is the result of an application of Magma procedure Simplify [BC10] on the presentation of the computed finitely presented group in the process. The lists of relations belong are of one of finitely-presented groups (which depends on the family of ET map) with generated as follows
   
   1. \(G = \langle x_1, x_2 \rangle\),
   2. \(G' = \langle t_1, t_2, f \rangle\), where \(f\) is mapped to an involution (in quotients),
   3. \(Q = \langle y_1, y_2, y_3, y_4, r \rangle\), where \(r\) is mapped to an involution,
   4. \(T = \langle z, a, b, s \rangle\), where \(s\) is mapped to an involution.
   
   The presentation defines the edge-transitive group isomorphic to one of \(\mathrm{Aut}\,\mathbf{M}\) or \(\mathrm{Aut}^+\,\mathbf{M}\) (see Lines 3, 4) and the correspondence is as follows

8. The map section header, along with some combinatorial properties of the embedding such as: the tag if map is non-degenerate in the manner as introduced in [OPPT], or if map is polytopal, or polyhedral [MT]. A map is polytopal, if the boundary cycle of every its face is a simple cycle. The map is polyhedral, if its face-width [MT] is greater that \(2\) (see e.g. here). Note that if map is polyhedral, then it is by definition non-degenerate, while polytopality does not imply non-degeneracy. However, the list sometimes claims that a map is polyhedral and non-degenerate. I decided not to fix this inaccuracy, because this information is not misleading.
   The oriented map itself is defined as a triple \(D; R,L\), where \(R,L\in\mathrm{Sym}(D)\) and \(L^2 = \mathrm{id}\).
9. Rotation of \(M\)
10. Dart-reversing involution of \(M\)
11. The list of local types. Note that an edge-transitive map may not be vertex-regular (and hence, uniform). The list is expressed as the set of Schläffli symbols
12. The external symmetries line may contain the following tags: If map is reflexible, then there exists the bijective mapping (the permutation of \(D\)) between the triples \((D;R,L)\to(D;R^{-1},L^{-1})\). The reflexible map may be selfdual, if there is a permutation \((D;R,L)\to(D;RL,L)\).
    If the map is not reflexible, then it must be chiral. A chiral map may be either positively-selfdual or negatively-selfdual. The map is positively-selfdual if there is a permutation mapping \((D;R,L)\to(D;R^{-1}L,L)\). The map is negatively-selfdual if there is a permutation mapping \((D;R,L)\to(D;RL,L)\). Moreover, if a map is reflexible and selfdual, then both former two permutations exist.
13. The optional reference to a dual of a map. The format of the tag is described above (see Line 1). This line is not present if the map is selfdual (of any kind).
14. The information about transitive actions of \(\mathrm{Aut}\,\mathbf{M}\) or \(\mathrm{Aut}^+\,\mathbf{M}\). The latter fact is marked by the present of the character `+` in the corresponding tag. The map can be (orientably-) vertex-transitive (VT), arc-transitive (AT), or regular (REG). Actually, if the group of orientation-preserving automorphisms is transitive on the edges of the map, then tag ET+ appears. Note that the transitivity on the faces was not tested.
15. The graph section contains the information on the graph. The graph may be simple and/or bipartite.
16. The numerical invariants of the graph records the number of vertices (v), the number of edges (e) and the number of faces (f) of the particular map \(M\).
17. The degree sequence of the graph in the form increasing sequence of integers with repetitions written as powers.

Enumerations

Census

Here is the actual census in the form of zipped archive: etran-catalog-20200120.zip.

References

BEB

Hans Ulrich Besche, Bettina Eick, and Eamon O’Brien, The Small Groups Library, (archived from original), last seen Sep 1, 2019.

BC10

Wieb Bosma, John Cannon, Claus Fieker, and Allan Steel, Handbook of Magma functions, Edition 2.16, 2010.

GAP

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.9.2, 2018.

GW

Jack E. Graver and Mark E. Watkins, Locally finite, planar, edge-transitive graphs, Mem. Amer. Math. Soc. 126 (1997), no. 601, vi+75.

KS

Ján Karabáš and Mária Skyvová, Census of oriented edge-transitive maps. (incomplete draft).

MT

Bojan Mohar and Carsten Thomassen, Graphs on Surfaces, John Hopkins University Press, ISBN: 9780801866890, 2001.

OPPT

Alen Orbanić, Daniel Pellicer, Tomaž Pisanski, and Thomas W. Tucker, Edge-transitive maps of low genus, Ars Mathematica Contemporanea 4 (2011), 385–402.

STW

Jozef Širáň, Thomas W. Tucker, and Mark E. Watkins, Realizing finite edge‐transitive orientable maps, J. Graph Theory, 37(2001), Vol. 1, 1-34.

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