Edge-antimagic labelings of forests

Abstract

An (a, d)-edge-antimagic total labeling of a graph G(V, E) is a one-to-one map f from V(G) U E(G) onto the integers {1,2,..., [V(G)| + |E(G)|} such that the edge-weights w(uv) = f(u) + f(uv) + f(v), uv € E(G), form an arithmetic progression with initial term a and common difference d. Such a labeling is called super if it has the property that the vertex labels are the smallest possible. In this paper we examine the existence of super (a, d)-edge-antimagic total labelings of forests, in which every component is a pathlike tree. Indeed, we prove that such a labeling exists when the forest has an odd number of components.