On distance-balanced generalized Petersen graphs

A connected graph $G$ of diameter $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)\ge \mathrm{\ell }$ is $\mathrm{\ell }$-distance-balanced if $\mathrm{\mid }{W}_{xy}\mathrm{\mid }=\mathrm{\mid }{W}_{yx}\mathrm{\mid }$ for every $x,y\in V(G)$ with $d_G(x,y)=\mathrm{\ell }$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,k)$ is $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(GP(n,k))$-distance-balanced provided that $n$ is large enough relative to $k$. This partially solves a conjecture posed by Miklavič and Šparl \cite{Miklavic:2018}. We also determine $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(GP(n,k))$ when $n$ is large enough relative to $k$.