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A CLASSIFICATION OF PRIME-VALENT REGULAR CAYLEY MAPS ON ABELIAN, DIHEDRAL AND DICYCLIC GROUPS
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 Title & Authors
A CLASSIFICATION OF PRIME-VALENT REGULAR CAYLEY MAPS ON ABELIAN, DIHEDRAL AND DICYCLIC GROUPS
Kim, Dong-Seok; Kwon, Young-Soo; Lee, Jae-Un;
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 Abstract
A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.
 Keywords
Cayley map;regular embedding;
 Language
English
 Cited by
 References
1.
M. Conder, R. Jajcay, and T. Tucker, Regular t-balanced Cayley maps, J. Combin. Theory Ser. B 97 (2007), no. 3, 453–473. crossref(new window)

2.
J. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996.

3.
L. C. Grove, Classical Groups and Geometric Algebra, American Mathematical Society, GSM 39, 2001.

4.
R. Jajcay, Automorphism groups of Cayley maps, J. Combin. Theory Ser. B 59 (1993), no. 2, 297–310. crossref(new window)

5.
R. Jajcay and J. Siran, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (2002), no. 1-3, 167–179. crossref(new window)

6.
J. H. Kwak, Y. S. Kwon, and R. Feng, A classification of regular t-balanced Cayley maps on dihedral groups, European J. Combin. 27 (2006), no. 3, 382–393. crossref(new window)

7.
J. H. Kwak, Y. S. Kwon, and J. M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Combin. Theory Ser. B 98 (2008), no. 3, 585–598. crossref(new window)

8.
J. H. Kwak and J.-M. Oh, A classification of regular t-balanced Cayley maps on dicyclic groups, European J. Combin. 29 (2008), no. 5, 1151–1159. crossref(new window)

9.
R. Nedela, Regular maps-combinatorial objects relating different fields of mathematics, J. Korean Math. Soc. 38 (2001), no. 5, 1069–1105.

10.
B. Richter, J. Siran, R. Jajcay, T. Tucker, and M. Watkins, Cayley maps, J. Combin. Theory Ser. B 95 (2005), no. 2, 189–245. crossref(new window)

11.
J. J. Rotman, An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148. Springer-Verlag, New York, 1995.

12.
M. Skoviera and J. Siran, Regular maps from Cayley graphs. I. Balanced Cayley maps, Discrete Math. 109 (1992), no. 1-3, 265–276. crossref(new window)

13.
J. Siran and M. ˇSkoviera, Regular maps from Cayley graphs. II. Antibalanced Cayleymaps, Discrete Math. 124 (1994), no. 1-3, 179–191. crossref(new window)

14.
Y. Wang and R. Feng, Regular balanced Cayley maps for cyclic, dihedral and generalized quaternion groups, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 4, 773–778. crossref(new window)