On Super Edge-magic Total Labeling of Reflexive W-trees

ON SUPER EDGE-MAGIC TOTAL LABELING OF REFLEXIVE W-TREES Muhammad Imran, Mehar Ali Malik, M. Yasir Hayat Malik Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad, Pakistan. E-mail: {imrandhab, alies.camp, yasirmalikawan}@gmail.com Mathematics Subject Classification: 05C78 ABSTRACT. Kotzig and Rosa [17] defined a magic labeling λ on a graph G to be a bijective mapping that assigns the integers from 1 to p+q to all the vertices and edges such that the sums of the labels on an edge and its two endpoints is constant for each edge. Ringel and Lladó [20] redefined this type of labeling as edge-magic. Recently, Enomoto et al. [6] introduced the name super edge-magic for magic labelings defined by Kotzig and Rosa, with an additional property that the vertices receive the smallest labels. That is, λ(V (G)) = {1,2,3,...,p}. If the domain of a labeling λ is the set of all vertices and edges of the graph G, then such labeling is called total labeling. The labelings which we study in this paper, have another property that, the weight ω(xy) xy E(G), calculated as; ω(xy) = λ(x) + λ(y) + λ(xy), is equal to a fixed constant k, called the magic constant or sometimes the valence of λ. A graph is called super edge-magic total (SEMT) if it admits a super edge-magic total labeling. In this paper, we construct new families of trees (using w-trees [15]), referred as reflexive w-trees and prove that they are super edge magic total. It is a classical problem to construct new classes of super edge magic total graphs using old ones.