## 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.

Send comments to Bojan.Mohar@uni-lj.si

##### Revised: July 16, 2005.