Gaussian Prime Labeling of Some Product Graphs

Abstract

A graph G on vertices is said to have prime labeling if there exists a labeling from the vertices of G to the first n natural numbers such that any two adjacent vertices have relatively prime labels.Gaussian integers are the complex numbers of the form $a+bi$ where $a,b \in \mathscr{Z}$ and $i^2 = -1$ and it is denoted by $\mathscr{Z}[i]$. A Gaussian prime labeling on G is a bijection from the vertices of G to $[\psi_n]$, the set of the first n Gaussian integers in the spiral ordering such that if $uv \in E(G)$, then $\psi_{(u)}$ and $\psi_{(v)}$ are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of product graphs.