A note on prime labeling k-partite k-graphs

Abstract

A graph has a prime labeling if its vertices can be assigned distinct numbers from 1 to $\left|V\right|$ so that the vertices on each edge receive relatively prime labels. This definition can be extended naturally to hypergraphs, whose edges may contain more than two vertices, in the following way. A hypergraph has a prime labeling if its vertices can be assigned distinct numbers from 1 to $\left|V\right|$ so that the $\gcd$ of numbers within each edge is 1 (which is sensible since greatest common divisor is defined for sets of numbers).

We examine the problem of prime labeling complete $k$-partite $k$-uniform hypergraphs. We prove that if this type of hypergraph has enough vertices and every pod of vertices is large enough, then it does not have a prime labeling. We also prove, on the other hand, that if a pod of vertices is small enough, then it does have a prime labeling.