On $\{1,2\}$-distance-balancedness of generalized Petersen graphs

A connected graph $G$ of diameter ${\rm diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, 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 Miklavi\v{c} and \v{S}parl [Discrete Appl.\ Math.\ 244 (2018) 143--154].