On super vertex magic total labeling of generalized petersen graphs

Abstract

Let G = (V(G),E(G)) be a finite simple graph with p - \V(G)\ vertices and q = \E(G)\ edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection / from V(G)UE(G) to the set of consecutive integers {1,2, ... ,p + q} with the property that, for every vertex u in V(G), the weight f(u) + £UV€£(C) f(uv) is a constant k. Moreover if f(V) = {1,2, ..., p}, f is called a super vertex magic total labeling. A graph is (super) vertex magic if it admits a (super) vertex magic total labeling. In 2002 MacDougall et al. first introduced the concept of vertex magic total labeling and studied their properties. In this paper we study the existence of the super vertex magic total labeling of disjoint unions of multiple copies of (not necessarily isomorphic) generalized Petersen graphs. We furthermore provide with more graphs admitting such labelings based upon adding arbitrary even regular factors to the above disjoint union of generalized Petersen graphs. Applications to other classes of graphs and open problems are included. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.