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

Abstract

A connected graph \(G = (V, E)\) is said to be \((a, d)\)-antimagic, for some positive integers \(a\) and \(d\), if its edges admit a labeling by all the integers in the set \(\{1, 2, \ldots, |E(G)|\}\) such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term \(a\) and the common difference \(d\). Mirka Miller and Martin Ba\'{e}a proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+3}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). The first author of this paper proved that \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 6\). In this paper, we show that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2} , 3)\)-antimagic for \(n \equiv 2 \pmod{4}\), \(n \geq 10\).