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 \(f^{+}:E(G)\to N\) of the edges of \(G\) defined by \(f^{+}(uv)=\gcd(f(u),f(v))\), \(\forall\) \(e=uv \in E(G)\) 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 \(mC_{n} \odot K_{1}\) which admit vertex $k$-prime labeling.