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

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