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

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, . . . , |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 Bac̆a proved that the generalized Petersen graph P (n, 2) is ( 3n+6 2 , 3)-antimagic for n ≡ 0 (mod 4), n ≥ 8 and conjectured that P (n, k) is ( 3n+6 2 , 3)antimagic for even n and 2 ≤ k ≤ n 2 − 1. The first author of this paper proved that P (n, 3) is ( 3n+6 2 , 3)-antimagic for even n ≥ 6. In this paper, we show that the generalized Petersen graph P (n, 2) is ( 3n+6 2 , 3)-antimagic for n ≡ 2 (mod 4), n ≥ 10.