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

A connected graph G of diameter diam(G) ≥ is -distance-balanced if |Wxy|=|Wyx| for every x,y∈ V(G) with dG(x,y)=, where Wxy is the set of vertices of G that are closer to x than to y. It is proved that if k≥ 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≥ 6, where k is even, and n>5/4k²+2k, or if k≥ 5, where k is odd, and n>7/4k²+3/4k, then $GP(n,k)$ is not $2$-distance-balanced. These results partially resolve a conjecture of Miklavi{c} and {S}parl [Discrete Appl.\ Math.\ 244 (2018) 143--154].