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Rotary hypermaps of genus 2. (English) Zbl 0968.05023

A hypermap on an orientable surface is called rotary if its orientation-preserving automorphism group acts transitively on the set of darts (hyperedge-vertex incidence pairs). A hypermap on any surface is called reflexible if its full automorphism group, including orientation-reversing automorphisms, acts transitively on the set of blades (hyperedge-vertex-face incidence triples).
The rotary hypermaps on the sphere and torus were classified in D. Corn and D. Singerman [Regular hypermaps, Eur. J. Comb. 9, No. 4, 337-351 (1988; Zbl 0665.57002)]. The article under review classifies the rotary hypermaps of genus 2. There are 43 of them, of which 10 are maps, classified in W. Threlfall [Gruppenbilder, Abh. Sächs. Akad. Wiss., Math.-Phys. Kl. 41, No. 6, 1-59 (1932; Zbl 0004.02202)], 20 more are obtained from the maps by considering them as hypermaps and applying hyperedge-vertex-face duality as discussed in A. Machì [On the complexity of a hypermap, Discrete Math. 42, 221-226 (1982; Zbl 0503.05024)], and the remaining 13 are obtained from the maps using first the bijection between bipartite maps and hypermaps of the same orientable genus presented in T. R. S. Walsh [Hypermaps versus bipartite maps, J. Comb. Theory, Ser. B 18, 155-163 (1975; Zbl 0302.05101)] and then hyperedge-vertex-face duality.
All these maps turn out to be reflexible. From this fact the authors deduce that there are no reflexible hypermaps on a (non-orientable) surface of characteristic \(-1\), generalizing an analogous result for maps in H. S. M. Coxeter and W. O. J. Moser [Generators and relations for discrete groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 14. Berlin-Heidelberg-New York: Springer-Verlag) (1972; Zbl 0239.20040)].

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C65 Hypergraphs
57M15 Relations of low-dimensional topology with graph theory
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