Edge-Antimagic Total Labeling of Disjoint Union of Caterpillars

Abstract

Let $G=(V,E)$ be a finite graph, where $V(G)$ and $E(G)$ are the (non-empty) sets of vertices and edges of $G$. An $(a,d)$-$edge-antimagictotallabeling$ is a bijection $\beta$ from $V(G)\cup E(G)$ to the set of consecutive integers $\{1,2,\dots ,|V(G)|+|E(G)|\}$ with the property that the set of all the edge-weights, $w(uv)=\beta (u)+\beta (uv)+\beta (v)$, for $uv\in E(G)$, is $\{a,a+d,a+2d,\dots ,a+(|E(G)|–1)d\}$, for two fixed integers $a>0$ and $d\ge 0$. Such a labeling is super if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super $(a,d)$-edge-antimagic total labelings for disjoint unions of multiple copies of a regular caterpillar.