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

Abstract: A connected graph $\Gamma$ of diameter ${\rm diam}(\Gamma) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(\Gamma)|=|W_{yx}(\Gamma)|$ for every $x,y\in V(\Gamma)$ with $d_{\Gamma}(x,y)=\ell$, where $W_{xy}(\Gamma)$ is the set of vertices of $\Gamma$ that are closer to $x$ than to $y$. $\Gamma$ is said to be highly distance-balanced if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(\Gamma)]$. 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 Miklavi\v{c} and \v{S}parl.