A note on V4 magic labelings of graphs. Issue 6 (1st November 2016) - Vandana, P. T.; Kumar, V. Anil
Abstract: For any abelian group ( A, +) with the identity element 0, we denote A * = A \{0}. Let G be a graph with a vertex set V ( G ) and an edge set E ( G ). Any mapping is called an edge labeling. If is an edge labeling, then we can define an induced vertex labeling as follows: A graph G is known as A -magic if there exists a labeling such that the induced vertex labeling is a constant map. That is, there exists an element k ∈ A such that for all . If k = 0, we say that the graph G is zero sum A -magic, otherwise G is k -sum A -magic. Observe that there exist graphs which are both zero sum and k -sum A magic. The Klein 4 group V 4 is the direct sum ℤ2 ⊕ ℤ2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. For simplicity we denote (0, 0), (0, 1), (1, 0), (1, 1) by 0, a, b, c respectively. In this paper we prove that the graph, the spanning subgraph of Kn ( n ≥ 4) with = E ( K n ) \ { v i v j :1 ≤ i < j ≤ r}, 2 ≤ r ≤ n − 2 is a -sum V 4 magic if and only if n is even and zero sum V 4 magic for all n . We also investigated a -sum V 4 magic labeling as well as the zero sum V 4 magic labeling of . We construct a new graph from by attaching r edges at the first r vertices v 1, v 2,  … , v r and n – r triangles at the remaining vertices of . Finally, we prove that is a -sum V magic if and only if n and r are of the same parity.