Non-$\ell$-distance-balanced generalized Petersen graphs $GP(n,3)$ and $GP(n,4)$

Abstract

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 $y$. We prove that the generalized Petersen graph $GP(n,3)$ where $n>16$ is not $\ell$-distance-balanced for any $1\le \ell < {\rm diam}(GP(n,3))$, and $GP(n,4)$ where $n>24$ is not $\ell$-distance-balanced for any $1\le \ell < {\rm diam}(GP(n,4))$. This partially solves a conjecture posed by \v{S}. Miklavi\v{c} and P. \v{S}parl (Discrete Appl. Math. 244:143-154, 2018).