Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D-2n

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 vertical bar a(n)=b(2)=1 >, S be a generating set of G, vertical bar S vertical bar=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=4s, and X similar or equal to Cay(G, {a, a(-1), a(i)b, a(-i)b}), where i(2)equivalent to +/- 1 (mod 2s), 2 <= i <= 2s-2. (C) 2005 Elsevier Ltd. All rights reserved.MathematicsSCI(E)26ARTICLE5750-7662