On vertex magic total labeling of disjoint union of sun 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) ∪ E(G) to the consecutive integers 1,2,...... ,p + q, with the property that, for every vertex u in V(G), one has f{u) + σuvϵE(G) f(uv) = k for some constant k. In 2004 MacDougall et al. first introduced the concept of vertex magic total labeling and studied their properties. In 2006 Slamin et al. studied such vertex magic total labeling of disconnected graphs. In this paper we study the properties of such vertex magic total labeling for various graph classes. Among others we settle a conjecture raised by Slamin et al. in 2006, which claims the existence of the vertex magic total labeling of disjoint union of multiple copies of distinct sun graphs, where the sun graph is the corona product of a cycle with a point. We furthermore provide with an infinite class of graphs admitting such labelings based upon adding arbitrary 4fc-regular factors to the above disjoint union of sun graphs. Note that the results we obtain in this paper could be extended to those pseudo-graphs with multiple edges or loops.