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

TL;DR

This paper extends the understanding of 4-valent Cayley graphs on dihedral groups by analyzing their structure and automorphism groups for larger generating sets, revealing their decomposition into circulant graphs and perfect matchings.

Contribution

It provides new structural and automorphism group results for Cayley graphs on dihedral groups with larger generating sets, building on prior work limited to smaller subsets.

Findings

Automorphism groups determined for dihedral groups of order 2p with rotation-only sets.

Cayley graphs with certain rotation sets decompose into two isomorphic circulant graphs.

Graphs generated by multiple reflections are bipartite and consist of disjoint perfect matchings.

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.