Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D2nD2n

A Cayley graph X=Cay(G,S) of group GG is said to be normal if R(G)R(G) is normal in Aut(X). Let G=〈a,b∣an=b2=1〉G=〈a,b∣an=b2=1〉, SS be a generating set of GG, |S|=4|S|=4. In this paper we show that any one-regular and 4-valent Cayley graph X=Cay(G,S) of dihedral groups GG is normal except that n=4sn=4s, and X≃Cay(G,{a,a−1,aib,a−ib}), where i2≡±1(mod2s)i2≡±1(mod2s), 2≤i≤2s−22≤i≤2s−2.