Neighborhood-Prime Labeling of some Generalized Petersen Graphs

Abstract

Let $G=(V(G),E(G))$ be a graph with $n$ vertices and for $v \in V(G)$, let $N(v)$ denote the open neighborhood of $v$. A bijective function $ f:V(G)\to \left\{1, 2, 3, \dots ,n\right\}$ is said to be a neighborhood-prime labeling of $G$, if for every vertex $v \in V(G)$ with $deg (v) > 1$, $gcd\left\{f(u): u\in N(v)\right\}=1.$ A graph which admits neighborhood-prime labeling is called a neighborhood-prime graph and if in a graph $G,$ every vertex is of degree at most $1,$ then such a graph is neighborhood-prime vacuously. In this paper, we show that the generalized Petersen graph $P(n,k)$ is neighborhood-prime when the greatest common divisor of $n$ and $k$ is $1, 2$ or $4$ and we also show that $P(n,8)$ is neighborhood-prime for all $n$.