On dihedrants admitting arc-regular group actions |
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Authors: | István Kovács Dragan Marušič Mikhail E Muzychuk |
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Institution: | (1) Department of Mathematics, University of Auckland, Private Bag, 92019 Auckland, New Zealand;(2) Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA;(3) Mathematics Department, Colgate University, Hamilton, NY 13346, USA |
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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 \((K_{n_{1}}\otimes\cdots\otimes K_{n_{\ell}})K_{m}^{\mathrm{c}}]\), where 2n=n 1???n ? m, 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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