On the super edge-magicness and the deficiency of some families of acyclic graphs

Abstract

Let G - (V, E) be a finite, simple and undirected graph having order v = |V(G) | and size e = |E(G)|. An edge magic total labeling of a graph G is a one-to-one map f from V(G) ∪ E(G) into the set of integers {1,2,...,v + e) with the property that, there is an integer constant c such that f(x) + f(xy) + f(y) = c, for any xy ∈ E(G). If f(V(G)) = {1,2,..., v} then an edge-magic total labeling is called a super edge-magic total labeling. The super edge-magic deficiency of a graph G, denoted by μ_s(G), is the minimum non-negative integer n such that G ∪ nK₁ has a super edge-magic total labeling or +∞ if there exists no such n. In this paper we find super edge-magic total labelings and the deficiency for forests consisting of extended w-trees, combs, stars and paths..