05023201

j

05023201 Wang, Changqun Xu, Mingyao Non-normal one-regular and 4-valent Cayley graphs of dihedral groups $D_{2n}$. Eur. J. Comb. 27, No. 5, 750-766 (2006). 2006 Elsevier Science (Academic Press), London EN

doi:10.1016/j.ejc.2004.12.007

Summary: A Cayley graph $X = \text{Cay}(G, S)$ of a group $G$ is said to be normal if $R(G)$ is normal in $\Aut(X)$. Let $G = \langle a, b\mid a^n = n^2 = 1, a^b=a^{-1}\rangle$, $S$ be a generating set of $G$, $|S| = 4$. In this paper we show that any one-regular and 4-valent Cayley graph $X = \text{Cay}(G, S)$ of dihedral group $G$ is normal except that $n = 4s$, and $X \simeq \text{Cay}(G, \{a, a^{-1}, a^ib, a^{-i}\})$, where $i^2 \equiv \pm 1 \pmod {2s}$, $2\leq i\leq 2s-2$.