On $\{1,2\}$-distance-balancedness of 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 $y$. It is proved that if $k\ge 3$ and $n>k(k+2)$, then the generalized Petersen graph $GP(n,k)$ is not distance-balanced and that $GP(k(k+2),k)$ is distance-balanced. This significantly improves the main result of Yang et al.\ [Electron.\ J.\ Combin.\ 16 (2009) #N33]. It is also proved that if $k\ge 6$, where $k$ is even, and $n>\frac{5}{4}{k}^2+2k$, or if $k\ge 5$, where $k$ is odd, and $n>\frac{7}{4}{k}^2+\frac{3}{4}k$, then $GP(n,k)$ is not $2$-distance-balanced. These results partially resolve a conjecture of Miklavič and Šparl [Discrete Appl.\ Math.\ 244 (2018) 143–154].