-labelings of Hamming graphs

Given integers c ≥ 0 and h ≥ k ≥ 1 , a c - L ( h , k ) -labeling of a graph G is a mapping f : V ( G ) → { 0 , 1 , 2 , … , c } such that | f ( u ) − f ( v ) | ≥ h if d G ( u , v ) = 1 and | f ( u ) − f ( v ) | ≥ k if d G ( u , v ) = 2 . The L ( h , k ) -number λ h , k ( G ) of G is the minimum c such that G has a c - L ( h , k ) -labeling. The Hamming graph is the Cartesian product of complete graphs. In this paper, we study L ( h , k ) -labeling numbers of Hamming graphs. In particular, we determine λ h , k ( K n q ) for 2 ≤ q ≤ p with h / k ≤ n − q + 1 or 2 ≤ q ≤ p with h / k ≥ q n − 2 q + 2 or q = p + 1 with h / k ≤ n / p , where p is the minimum prime factor of n .