A note on prime labeling k-partite k-graphs

A graph has a prime labeling if its vertices can be assigned distinct numbers from 1 to vertical bar V vertical bar 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 vertical bar V 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.