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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-antimagic total labeling is a bijection β from V(G) U E(G) to the set of consecutive integers {1, 2,..., V(G) + E(G) } with the property that the set of all the edge-weights, w(w) = β(u) + β(w) + β(v), uv ∈ E(G), is {a, a + d, a + 2d,..., a + ( E(G) - 1)d}, for two fixed integers a > 0 and d > 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 union of multiple copies of a regular caterpillar.