On dihedrants admitting arc-regular group actions
István Kovács
, Dragan Marušič
and Mikhail E. Muzychuk
and Dragan Maru {s}i\check c was supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije,” research program P1-0285. I. Kovács \cdot D. Maru {s}i\check c ( ) FAMNIT, University of Primorska, Glagolja {s}ka 8, 6000 Koper, Slovenia
DOI: 10.1007/s10801-010-0251-7
Abstract
We consider Cayley graphs Γ over dihedral groups, dihedrants for short, which admit an automorphism group G acting regularly on the arc set of Γ . We prove that, if D 2 n \leq G\leq Aut( Γ ) is a regular dihedral subgroup of G of order 2 n such that any of its index 2 cyclic subgroups is core-free in G, then Γ belongs to the family of graphs of the form ( K n 1 Ä \frac{1}{4} Ä K n l)[ K m c] (K_{n_{1}}\otimes\cdots\otimes K_{n_{\ell}})[K_{m}^{\mathrm{c}}], where 2 n= n 1\cdot \cdot \cdot n \ell m, and the numbers n i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.
Pages: 409–426
Keywords: keywords arc-transitive graph; Cayley graph; Cayley map; dihedral group; core-free group
Full Text: PDF
References
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2. Conder, M., Jajcay, R., Tucker, T.: Regular t -balanced Cayley maps. J. Combin. Theory, Ser. B 97, 453-473 (2007)
3. Conder, M., Jajcay, R., Tucker, T.: Regular Cayley maps for finite abelian groups. J. Algebr. Comb. 25, 259-283 (2007)
4. Du, S.F., Malni\check c, A., Maruši\check c, D.: Classification of 2-arc-transitive dihedrants. J. Combin. Theory, Ser. B 98, 1349-1372 (2008)
5. Feng, R., Wang, Y.: Regular balanced Cayley maps for cyclic, dihedral and generalized quaternion groups. Acta Math. Sinica (Engl. Ser.) 21, 773-778 (2005)
6. Herzog, M., Kaplan, G.: Large cyclic subgroups contain non-trivial normal subgroups. J. Group Theory 4, 247-253 (2001)
7. Kim, D., Kwon, Y.S., Lee, J.: Classification of p-valent regular Cayley maps on dihedral groups, manuscript
8. Kovács, I.: Classifying arc-transitive circulants. J. Algebr. Comb. 20, 353-358 (2004)
9. Kwak, Y.H., Kwon, Y.S., Feng, R.: Classification of t -balanced Cayley maps on dihedral groups. Eur. J. Comb. 27, 382-393 (2006)
10. Kwak, Y.H., Oh, Y.M.: One-regular normal Cayley graphs on dihedral groups of valency 4 or 6 with cyclic vertex stabilizer. Acta Math. Sin. (Engl. Ser.) 22, 1305-1320 (2006)
11. Kwak, Y.H., Kwon, Y.S., Oh, Y.-M.: Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency. J. Comb. Theory, Ser. B 98, 585-598 (2008)
12. Li, C.H.: Permutation groups with a cyclic regular subgroup and arc-transitive circulants. J. Algebr. Comb. 21, 131-136 (2005)
13. Li, C.H.: Finite edge-transitive Cayley graphs and rotary Cayley maps. Trans. Am. Math. Soc. 358, 4605-4635 (2006)
14. Li, C.H., Finite, J. Pan: 2-arc transitive abelian Cayley graphs. Eur. J. Comb. 29, 148-158 (2007)
15. Maruši\check c, D.: On 2-arc-transitivity of Cayley graphs. J. Comb. Theory, Ser. B 87, 162-196 (2003)
16. Maruši\check c, D.: Corrigendum to “On 2-arc-transitivity of Cayley graphs”. J. Comb. Theory, Ser. B 96, 761-764 (2006) [J. Comb. Theory, Ser. B 87, 162-196 (2003)]
17. McSorley, J.P.: Cyclic permutations in doubly-transitive groups. Commun. Algebra 25, 33-35 (1997)
18. Muzychuk, M.: On the structure of basic sets of Schur rings over cyclic groups. J. Algebra 169(2), 655-678 (1994) J Algebr Comb (2011) 33: 409-426
19. Muzychuk, M.: On the isomorphism problem for cyclic combinatorial objects. Discrete Math. 197/198, 589-605 (1999)
20. Nedela, R.: Regular maps-combinatorial objects relating different fields of mathematics. J. Korean Math. Soc. 38, 1069-1105 (2001)
21. Richter, B., Širá\check n, J., Jajcay, R., Tucker, T., Watkins, M.: Cayley maps. J. Combin. Theory, Ser. B 95, 189-245 (2005)
22. Wang, C.Q., Xu, M.Y.: Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D2n. Eur. J. Comb. 27, 750-766 (2006)
23. Wang, C.Q., Zhou, Z.Y.: One-regularity of 4-valent and normal Cayley graphs of dihedral groups D2n.
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