Edge-Antimagic Total Labeling of Disjoint Union of Caterpillars

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 \((a, d)\)-\emph{edge-antimagic total labeling} 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 \( d \geq 0 \). 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.