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?

[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