20.500.12556/RUL-175773 The conjecture on distance-balancedness of generalized Petersen graphs holds when internal edges have jumps $3$ or $4$ A connected graph $G$ with ${\rm diam}(G) \ge \ell$ 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 $y$. Miklavič and Šparl [Discrete Appl. Math. 244 (2018), 143--154] conjectured that if $n > n_k$ where where $n_k = 11$ if $k = 2$, $n_k = (k+1)^2$ if $k$ is odd, and $n_k = k(k +2)$ if $k \ge 4$ is even, then the generalized Petersen graph $GP(n,k)$ is not $\ell$-distance-balanced for any $1\le \ell<{\rm diam}(GP(n,k))$. In the seminal paper, the conjecture was verified for $k=2$. In this paper we prove that the conjecture holds for $k=3$ and for $k=4$. distance-balanced graph $\ell$-distance-balanced graph generalized Petersen graph diameter razdaljno uravnoteženi graf $\ell$-razdaljno uravnoteženi graf posplošeni Petersenov graf premer true false true Angleški jezik Slovenski jezik Članek v reviji 2025-11-07 09:01:14 2025-11-07 09:01:17 2025-12-15 03:46:58 0000-00-00 00:00:00 2025 0 0 Str. 703-733 no. 2 Vol. 68 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 1669-9637 10.33044/revuma.4824 256284675 v68n2a18.pdf v68n2a18.pdf 1 AE8D0C62D38C3C1673E852555A36CA47 db90a687fdde42bc3e90c15331b1db444c7faec46f86ca4f9967232bc16e37aa 71f028c9-bbb1-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=221624 https://inmabb.criba.edu.ar/revuma/revuma.php?p=doi/v68n2a18 1 a089557f-bbaf-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=221622 Fakulteta za matematiko in fiziko 0 0 0