Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups

Abstract

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$. Kaseasbeh and Erfanian (2021) determined the structure of all Cayley graphs on the dihedral group of order $2n$ for subsets $S$ of size at most three. We extend their work by analyzing the structure of such Cayley graphs for subsets $S$ of size at least four. Our main results are as follows: 1. using a classical result of Burnside and Schur, we determine the automorphism groups of Cayley graphs on dihedral groups of order $2p$, where $p$ ranges over infinitely many primes and $S$ consists only of rotations; 2. if $S$ consists of $4 \le 2k < n$ distinct rotations, then the Cayley graph $\mathrm{Cay}(D_{2n},S)$ is the disjoint union of two isomorphic circulant graphs on $n$ vertices, and 3. if $S$ is a generating set of $4\leq k\leq n$ reflections, then the Cayley graph $\mathrm{Cay}(D_{2n},S)$ is bipartite, forming the disjoint union of $k$ perfect matchings.