Some Distance Antimagic Labeled Graphs

Abstract

Let G be a graph of order n. A bijection $f:V(G) \longrightarrow \{1, 2, \ldots , n\}$ is said to be distance antimagic if for every vertex v the vertex weight defined by $w_f(v) =\sum _{x\in N(v)}f(x)$ is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define $f_{k}:V(G) \longrightarrow \{1+k,2+k, \ldots , n+k\}$ by $f_{k}(x)=f(x)+k$. If $w_{f_{k}}(u)\ne \ w_{f_{k}}(v)$ for every pair of vertices $u,v \in V$, for any $k\ge 0$ 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 $rP_n$, generalised Petersen graph $P(n,k), \ n \ge 5$, Harary graph $H_{4,n}$ for $n\ne 6$ and also prove that join of these graphs is distance antimagic.

A.K. Handa—Also senior lecturer in mathematics at Padre Conceicao College of Engineering.