Prime labelings of generalized Petersen graphs

Abstract

A graph $G$ is called prime if the vertices of $G$ can be assigned distinct labels $1,2,\dots ,\left|V\left(G\right)\right|$ such that the labels on any two adjacent vertices are relatively prime. By showing that for every even $n\le 2.468\times 10^9$ there exists $s\in \left[1,n-1\right]$ such that both $n+s$ and $2n+s$ are prime, we prove the generalized Peterson graph $P\left(n,1\right)$ is prime for all even $n\in \left[4,2.468\times 10^9\right]$. Moreover, for a fixed $n$ we describe a method for labeling $P\left(n,k\right)$ that is a prime labeling for multiple values of $k$. Using this method, we prove $P\left(n,k\right)$ is prime for all even $n\le 50$ and all odd $k\in \left[1,n∕2\right)$.