On dihedrants admitting arc-regular group actions

István Kovács; Dragan Marušič; Mikhail E. Muzychuk

doi:10.1007/s10801-010-0251-7

On dihedrants admitting arc-regular group actions

István Kovács, Dragan Marušič, Mikhail E. Muzychuk

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

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 _2n ≤ G≤ Aut(Γ) is a regular dihedral subgroup of G of order 2n 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_n1⊗⋯⊗ K_nl)[K_m ^c], where 2n=n ₁⋯n_lm, and the numbers n _i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.

Original language	English
Pages (from-to)	409-426
Number of pages	18
Journal	Journal of Algebraic Combinatorics
Volume	33
Issue number	3
DOIs	https://doi.org/10.1007/s10801-010-0251-7
State	Published - 1 May 2011
Externally published	Yes

Keywords

Arc-transitive graph
Cayley graph
Cayley map
Core-free group
Dihedral group

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-010-0251-7

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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 2n ≤ G≤ Aut(Γ) is a regular dihedral subgroup of G of order 2n 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 (Kn1⊗⋯⊗ Knl)[Km c], where 2n=n 1⋯nlm, and the numbers n i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.",

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AU - Kovács, István

AU - Marušič, Dragan

AU - Muzychuk, Mikhail E.

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N2 - 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 2n ≤ G≤ Aut(Γ) is a regular dihedral subgroup of G of order 2n 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 (Kn1⊗⋯⊗ Knl)[Km c], where 2n=n 1⋯nlm, and the numbers n i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.

AB - 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 2n ≤ G≤ Aut(Γ) is a regular dihedral subgroup of G of order 2n 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 (Kn1⊗⋯⊗ Knl)[Km c], where 2n=n 1⋯nlm, and the numbers n i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.

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KW - Cayley map

KW - Core-free group

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On dihedrants admitting arc-regular group actions

Abstract

Keywords

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