On $\ell$-distance-balancedness of cubic Cayley graphs of dihedral groups

Abstract

A connected graph $Γ$ of diameter ${\rm diam}(Γ) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(Γ)|=|W_{yx}(Γ)|$ for every $x,y\in V(Γ)$ with $d_Γ(x,y)=\ell$, where $W_{xy}(Γ)$ is the set of vertices of $Γ$ that are closer to $x$ than to $y$. $Γ$ is said to be highly distance-balanced if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(Γ)]$. It is proved that every cubic Cayley graph whose generating set is one of $\{a,a^{n-1},ba^r\}$ and $\{a^k,a^{n-k},ba^t\}$ is highly distance-balanced. This partially solves a problem posed by Miklavič and Šparl.