On Super $a,d$-edge Antimagic Total Labeling of Disconnected Graphs

Abstract

A graph $G$ is called $(a,d)$-\text{edge antimagic total} $(a,d)$-\text{EAT} if there exist integers $a>0,d\ge 0$ and a bijection $\lambda :V\cup E\to \{1,2,\dots ,|V|+|E|\}$ such that

$$W=\{w(xy):xy\in E\}=\{a,a+d,\dots ,a+(|E|-1)d\},$$

where $w(xy)=\lambda (x)+\lambda (y)+\lambda (xy)$. An $(a,d)$-EAT labeling $\lambda$ of graph $G$ is \text{super} if $\lambda (V)=\{1,2,\dots ,|V|\}$. In this paper, we describe how to construct a super $(a,d)$-EAT labeling on some classes of disconnected graphs, namely $P_n\cup P_{n+1}$, $n{P}_2\cup P_n$, and $n{P}_2\cup P_{n+2}$, for positive integer $n$.