A Study On $(a,d)$-Antimagic Graphs Using Partition

Abstract

A connected graph $G(V,E)$ is said to be $(a,d)$-antimagic if there exist positive integers $a$ and $d$ and a bijection $f:E\to \{1,2,\dots ,|E|\}$ such that the induced mapping ${\text{g}}_{\text{f}}:V\to \mathbb{N}$ defined by ${\text{g}}_{\text{f}}(v)=\sum _{\text{e}\in \text{I}(v)}\text{f(e)}$, where $\text{I}(v)=\{\text{e}\in E\mid \text{e}\text{ is incident to }v\}$, $v\in V$ is injective and ${\text{g}}_{\text{f}}(V)=\{a,a+d,a+2d,\dots ,a+(|V|-1)d\}$. In this paper, using partition, we prove that (i) the 1-sided infinite path $P_1$ is $(1,2)$-antimagic, (ii) the path $P_{2n+1}$ is $(n,1)$-antimagic, and (iii) the $(n+2,1)$-antimagic labeling is the unique $(a,d)$-antimagic labeling of $C_{2n+1}$; and the graphs $K_1+(K_1\cup K_2)$, $P_{2n}$, and $C_{2n}$ are not $(a,d)$-antimagic. For $a,d\in \mathbb{N}$, on an $(a,d)$-antimagic graph $G$, we obtain a new relation, $a+(p-1)d\le \frac{\mathrm{\Delta }(2q–\mathrm{\Delta }+1)}{2}$. Using the results on $(a,d)$-antimagic labeling of $C_{2n}$ and $C_{2n+1}$, we obtain results on the existence of $(a,d)$-arithmetic sequences of length $2n$ and $2n+1$, respectively.