$(a,d)$-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 $\{1,2,\dots ,|V|+|E|\}$ such that the set $\{f(v)+\sum f(uv)\mid uv\in E\}$, where $v$ ranges over all of $V$, is $\{a,a+d,a+2d,\dots ,a+(|V|-1)d\}$. 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.