Regular Maps on Nonorientable Surfaces

A map on a surface is *regular* if the order of the automorphism group of the map is *2 | E(G) |* for oriented surfaces and *4 | E(G) |* for nonorientable surfaces. Regular maps have the largest possible automorphism groups over all embeddings of a graph (in the orientable case the embedding must preserve an orientation of the surface).

The problem is to construct a regular map for each surface. This is easy in the orientable case: the bouquet of *2g* circles has a regular embedding on the sphere with *g* handles attached (folklore, cf. [SS]). The nonorientable case is harder. *K_6* has a regular map on the projective plane. However, Coxeter and Moser show that the Klein bottle and the sphere with three crosscaps do not admit a regular map. Jozef Siran relays that Steve Wilson might have some other nonexistence results. On the positive side, he says that M. Condor, M. Skoviera, and R. Nedela have constructed regular maps for roughly two-thirds of the nonorientable surfaces. Condor and Everitt [CE] have raised this to 77% of the surfaces.

**Conjecture:** *For some* c, *every surface of nonorientable genus at least* c *admits a regular map.*

I am unsure of the origin of the above conjecture, but I believe it to be true.

References:

[CE] M. Condor and B. Everitt, Regular maps on nonorientable surfaces, *Geom. Dedicata* **56** (1995) 209-219.

[CM] Coxeter and Moser, Generators and relations for discrete groups.

[SS] J. Siran and M. Skoviera, Regular maps from Cayley Graphs, Part 1: Balanced Cayley maps, *Discrete Math.* **109** (1992) 265-276.

Submitted by: Dan Archdeacon, Dept. of Math. and Stat., University of Vermont, Burlington VT, USA 05405

Send comments to *dan.archdeacon@uvm.edu *

August, 1995. Updated September, 1996