VERTEX-MAGIC TOTAL LABELINGS OF DISCONNECTED GRAPHS

Abstract

Let G be a graph with vertex set V = V (G) and edge set
E = E(G) and let e = |E(G)| and v = |V (G)|. A one-to-one map λ from
V ∪ E onto the integers {1, 2, ..., v + e} is called vertex magic total labeling
if there is a constant k so that for every vertex x,
λ(x) + Xλ(xy) = k
where the sum is over all vertices y adjacent to x. Let us call the sum of
labels at vertex x the weight wλ(x) of the vertex under labeling λ; we require wλ(x) = k for all x. The constant k is called the magic constant for λ.
In this paper, we present the vertex magic total labelings of disconnected
graph, in particular, two copies of isomorphic generalized Petersen graphs
2P(n, m), disjoint union of two non-isomorphic suns Sm ∪ Sn and t copies
of isomorphic suns tSn.