On \((a, d)\)-Antimagic Labelings of Generalized Petersen Graphs \(P(n, 3)\)

Abstract

A connected graph \(G = (V, E)\) is said to be \((a,d)\)-antimagic if there exist positive integers \(a,d\) and a bijection \(f : E \to \{1,2,\ldots,|E|\}\) such that the induced mapping \(g_f : V \to \mathbb{N}\), defined by \(g_f(v) = \sum f(uv)\),\({uv \in E(G)}\) is injective and \(g_f(V) = \{a,a+d,\ldots,a+(|V|-1)d\}\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that the generalized Petersen graph \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 8\).