Prime labelings on planar grid graphs

A graph G is said to be prime if there is a bijective function f : V ( G ) → { 1 , 2 , . . . , | V ( G ) |} such that f ( u ) and f ( v ) are relatively prime whenever u is adjacent to v . It is known that for any prime p and any integer n such that 1 ≤ n ≤ p , there exists a prime labeling on the p × n planar grid graph P p × P n . We show that P p × P n has a prime labeling for any odd prime p and any integer n such that p < n ≤ p 2 . We discuss how this approach may lead to prime labeling on P p × P n for any odd prime p and any positive integer n .