Non-normal one-regular and 4-valent Cayley graphs of dihedral groups

A Cayley graph X = Cay ( G , S ) of group G is said to be normal if R ( G ) is normal in Aut ( X ) . Let G = 〈 a , b ∣ a n = b 2 = 1 〉 , S be a generating set of G , | S | = 4 . In this paper we show that any one-regular and 4-valent Cayley graph X = Cay ( G , S ) of dihedral groups G is normal except that n = 4 s , and X ≃ Cay ( G , { a , a − 1 , a i b , a − i b } ) , where i 2 ≡ ± 1 ( mod 2 s ) , 2 ≤ i ≤ 2 s − 2 .