Regular Cayley Maps that are Neither Balanced nor Antibalanced

The problem below has been solved by Jajcey and Siran. They showed the existence of regular Cayley maps that are neither balanced nor antibalanced. They report that in fact, maps of any generator type are possible.

Given a finite group *G*, a symmetric set of generators *X* of *G* (i.e. *x*^{-1} is in *X* whenever *x* is in *X*), and a cyclic permutation *p* on *X*, a *Cayley map* *CM(G,X,p)* is a 2-cell embedding of the *Cayley graph* *C(G,X)* into an orientable surface with the same local orientation *p* at every vertex. A *map-automorphism* *A* of a Cayley map *M = CM(G,X,p)* is an oriented-region-preserving permutation of the set of arcs of *M*. The group of all map-automorphisms of *M*, *AutM*, is always vertex-transitive thanks to the left-translation action of the underlying *G*. Also, the stabilizer of any of the arcs of *M* is known to be trivial [BW], and so *|AutM| <= |G| |X|*. The identity *|AutM| = |G| |X|* is equivalent to the arc-transitivity of *AutM*, and Cayley maps with arc-transitive automorphism groups are called *regular*.

A classical implication of Biggs and White [BW] asserts that the existence of a group-automorphism *rho* with the property *rho | X = p* yields the regularity of *CM(G,X,p)*. The reversed assertion has been shown to be true by Siran and Skoviera [SS1] for the special case of *balanced* Cayley maps satisfying the additional property *p(x^{-1}) = p(x)^{-1}*. The original implication cannot be, however, reversed in general, because of the existence of regular *antibalanced* Cayley maps [SS2] satisfying the property *p(x^{-1}) = (p^{-1}(x))^{-1}*. In 1993 a complete characterization of regular Cayley maps generalizing these results was found [J] in which the regularity of a Cayley map has been shown to be equivalent to the existence of a special graph-automorphism *rho* of the underlying Cayley graph that preserves the identity of the underlying group, is equal to the cyclic permutation *p* on *Omega* and satisfies the rather complex identity *p( rho(a)^{-1} rho(ax)) = rho(a)^{-1} rho(a p(x))*, for all *a* in *G*, *x* in *X*. While all the previously studied and well understood balanced and antibalanced cases certainly satisfy these conditions, the conditions do not seem to disqualify the possibility of the existence of a regular Cayley map that is neither balanced nor antibalanced. Nevertheless, to the author's best knowledge, there are no examples known of such ``strange'' regular Cayley maps.

**Question:** *Are there any regular Cayley maps that are neither balanced nor antibalanced?*

References:

[BW] N.Biggs and A.T.White, Permutation Groups and Combinatorial Structures, *Math. Soc. Lect. Notes* **33 **(Cambridge Univ. Press, Cambridge, 1979).

[J] R.Jajcay, Automorphism groups of Cayley maps, *J. of Combin. Th. Ser. B* **59** (1993) 297-310.

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

[SS2]M. Skoviera and J. Siran, Regular Maps from Cayley Graphs II. Antibalanced Cayley Maps, *Discrete Math.* **124** (1994) 179-191.

Submitted by: Robert Jajcay

Send comments to *dan.archdeacon@uvm.edu *and *jajcay@helios.unl.edu *

August, 1995. Modified November, 1998