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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title = "On dihedrants admitting arc-regular group actions",

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.",

keywords = "Arc-transitive graph, Cayley graph, Cayley map, Core-free group, Dihedral group",

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year = "2011",

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doi = "10.1007/s10801-010-0251-7",

language = "English",

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TY - JOUR

T1 - On dihedrants admitting arc-regular group actions

AU - Kovács, István

AU - Marušič, Dragan

AU - Muzychuk, Mikhail E.

PY - 2011/5/1

Y1 - 2011/5/1

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.

KW - Arc-transitive graph

KW - Cayley graph

KW - Cayley map

KW - Core-free group

KW - Dihedral group

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DO - 10.1007/s10801-010-0251-7

M3 - Article

AN - SCOPUS:79952449115

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SP - 409

EP - 426

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 3

ER -

On dihedrants admitting arc-regular group actions

Abstract

Keywords

ASJC Scopus subject areas

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