E-super vertex magic labeling for caterpillar graph

Abstract

Graph labeling has been an exciting area of research in graph theory. Most graph labeling origins can be traced back to the mid-1960s. Over the last 60 years, more than 100 graph-labelling techniques have been studied. E-super vertex magic labeling is a type of modern concept when compared with other popular labeling patterns such as graceful labeling, harmonious labeling, lucky labeling, anti-magic labeling, etc. It is one of the most challenging and interesting labeling techniques with various applications. For a finite simple graph G, the set of vertices and edges are denoted by V(G) and E(G), respectively. If G is a simple undirected graph with p vertices and q edges, then vertex magic total labeling is a bijective map f from V(G) ∪ E(G) onto the set { 1, 2, . . . , p + q, } with the property that, for every vertex u in V(G), f(u)+ ∑ V∈N(u)f (uv) = k, where k is a constant and set N(u) denotes the vertices adjacent to the vertex u. The labeling is called E-super vertex magic if f(E(G)) = {1, 2, ... , q}. A graph G is called E-super vertex magic if it admits an E-super vertex magic labeling. Intending to answer the open problem, every tree is E- super vertex magic to some extent; we focus on the E-super vertex magic labeling for the caterpillar graph. In this study, we can prove that E-super vertex magic labeling does not exist for the star graphs Sₙ with n ≥ 4. Consequently, we proved that the caterpillar graph, which has a star graph as a subgraph, is not an E- super vertex graph for order; n ≥ 7.