Vertex \(k\)-Prime Labeling of Cyclic Snakes

Abstract

. For each positive integer k , a simple graph G of order p is said to be k -prime labeling if there exists an injective function f whose labels are from k to k + p − 1 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 every pair of neighbouring vertices are relatively prime. This type of graph is known as a k -prime graph. In this paper, we redefine the labeling as vertex k -prime labeling for some k positive integers and study some cyclic snake graphs and corona graphs of the form mC n ⊙ K 1 which admit vertex k -prime labeling.