\((a, d)\)-Edge-Antimagic Total Labelings Of Cycle

Abstract

An \((a, d)\)-edge-antimagic total labeling for a graph \(G(V, E)\) is an injective mapping \(f\) from \(V \cup E\) onto the set \(\{1, 2, \ldots, |V| + |E|\}\) such that the set \(\{f(v) + \sum f(uv) \mid uv \in E\}\), where \(v\) ranges over all of \(V\), is \(\{a, a+d, a+2d, \ldots, a+(|V|-1)d\}\). Simanjuntak et al conjecture:1. \(C_{2n}\) has a \((2n + 3, 4)\)- or a \((2n + 4, 4)\)-edge-antimagic total labeling;
2. cycles have no \((a, d)\)-edge-antimagic total labelings with \(d > 5\).In this paper, these conjectures are shown to be true.