K - Prime Labeling of One Point Union of Path Graph

Abstract

Abstract A k – prime labeling of a graph G is an injective function f:V → {k, k+1, k+2, …,k + |V| − 1} for some positive integer k that induces a function f+ :E(G) → N of the edges of G defined by f+(uv) = gcd(f(u),f(v)), ∀ e = uv ∈ E(G) such that gcd(f(u),f(v)) = 1. A graph G that admits k – prime labeling is called a k – prime graph. In this paper, we have proved that the conditions of k - prime labeling are satisfied for Y – tree, X – tree and further extended our result to one point union of path graphs. We have also given an application of prime labeling and k – prime labeling for one point union of path graphs.