Some distance antimagic labeled graphs

Abstract

Let G be a graph of order n. A bijection f: V (G) → {1, 2,..., n} is said to be distance antimagic if for every vertex v the vertex weight defined by wf (v) = Σx∈N(v) f(x) is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define fk: V (G) −→ {1+k,2+k,..., n+k} by fk(x) = f(x) + k. If wfk (u) ≠ wfk (v) for every pair of vertices u, v ∈ V, for any k ≥ 0 then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. In this paper, we provide arbitrarily distance antimagic labelings for rPn, generalised Petersen graph P(n, k), n ≥ 5, Harary graph H4,n for n ≠ = 6 and also prove that join of these graphs is distance antimagic.