## Genus spectrum of orientably regular maps

A map M on an orientable surface S is orientably regular if the group of orientation-preserving automorphisms of the map acts transitively on the darts (pairs (v,e), where v is a vertex and e is an edge incident with v). A map is simple if its underlying graph and its dual graph are both simple graphs (loopless and without multiple edges). The genus spectrum of a family of maps is the set of all integers g for which there exists a map in the family whose genus is equal to g.

Problem 1: Determine the genus spectrum of all simple orientably regular maps.

This problem has been suggested by David Surowski and Marston Conder.

It is not known if the genus spectrum in Problem 1 contains all sufficiently large integers.

Problem 2: Determine the genus spectrum of all orientably regular triangulations.

Conder and Dobcsányi [1] have determined all (orientably) regular maps for surfaces of genus at most 15. Their results show that there are no regular maps with triangular faces on orientable surfaces of genera 11 and 12, but there are examples on all surfaces of genera 0 to 10 and 13 to 15.

Problem 3: Let k 3 be an integer. Determine the genus spectrum of all orientably regular maps whose face-width is at least k.

For k = 3, Problem 3 is related to Problem 2 since triangulations always have face-width at least 3. For large values of k, such maps will not exist for small values of the genus g (except for g = 0 and 1) since there are only finitely many (orientably) regular maps of a given genus g 2. But it is possible that they exist for all but finitely many values of g. Existence of regular maps of arbitrarily large face-width has been proved by Nedela and Škoviera [2].

Bibliography:

[1] M. Conder, P. Dobcsányi, Determination of all regular maps of small genus, J. Combin. Theory Ser. B 81 (2001), 224-242.

[2] R. Nedela, M. Škoviera, Regular maps on surfaces with large planar width, European J. Combin. 22 (2001), 243-261.