On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups

TL;DR

This paper characterizes when dihedral groups of order 2n are 4-DCI or 4-CI groups, extending previous results by identifying divisibility conditions on n for these properties.

Contribution

It establishes new criteria for dihedral groups to be 4-DCI or 4-CI groups based on the parity and divisibility of n.

Findings

G is a 4-DCI-group iff n is odd and not divisible by 9.

G is a 4-CI-group iff n is odd.

Previous results confirmed for m ≤ 3 are extended to m = 4.

Abstract

Let $m$ be a positive integer. A group $G$ is said to be an $m$-DCI-group or an $m$-CI-group if $G$ has the $k$-DCI property or $k$-CI property for all positive integers $k$ at most $m$, respectively. Let $G$ be a dihedral group of order $2n$ with $n\geq 3$. Qu and Yu proved that $G$ is an $m$-DCI-group or $m$-CI-group, for every $m\in \{1,2,3\}$, if and only if $n$ is odd. In this paper, it is shown that $G$ is a $4$-DCI-group if and only if $n$ is odd and not divisible by $9$, and $G$ is a $4$-CI-group if and only if $n$ is odd.