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M. Oh, Regular t-balanced Cayley maps on semi-dihedral groups, J. Combin. Theory Ser. B 99, no. 2, 480–493 (2009). 17. B. Richter, R. Jajcay, J. W. E. Watkins, Cayley maps, J. Combin. Theory Ser. B 95, 189–245 (2005). 18. M. Škoviera and J. Širáˇn, Regular Maps from Cayley Graphs, Part I. Balanced Cayley Maps, Discrete Math. 109, 265–276 (1992). 19. J. Širáˇn and M. Škoviera, Regular Maps from Cayley Graphs II. Antibalanced Cayley Maps, Discrete Math. 124, 179–191 (1994). 20. Y. -Q. Feng, Regular balanced Cayley maps for cyclic, dihedral and generalized quaternion groups, Acta Math.

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Then ϕ m+1 (1) = = ϕ(1 + (h + th + t 2 h + · · · + t m−1 h)) = ϕ((h + th + t 2 h + · · · + t m−1 h) + 1) = = ϕ(h + th + t 2 h + · · · + t m−1 h) + ϕ(1) = th + t 2 h + · · · + t m h + 1 + h = = 1 + h + th + t 2 h + · · · + t m h, which proves the induction claim. Plugging the parameters (n; b, t, h, p) into the equations (8) and (7) allows for a deterministic calculation of ϕ(a), for all a ∈ Zn . Thus, ϕ is uniquely determined by the parameters (n; b, t, h, p). Finally, if the five-tuples associated with two skew-morphisms ϕ, ϕ differ in at least one number, it is easy to see that the two skew-morphisms differ as permutations.

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