On \((a, d)\)-Antimagic Labelings of Generalized Petersen Graphs \(P(n, 2)\)

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 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 \(P(n, k)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for odd \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n,2)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for \(n \equiv 3 \pmod{4}\), \(n \geq 7\).