-labeling of the infinite regular trees

An L ( p , q , r ) -labeling of a graph G is defined as a function f from the vertex set V ( G ) into the nonnegative integers such that for any two vertices x , y , | f ( x ) − f ( y ) | ≥ p if d ( x , y ) = 1 , | f ( x ) − f ( y ) | ≥ q if d ( x , y ) = 2 and | f ( x ) − f ( y ) | ≥ r if d ( x , y ) = 3 , where d ( x , y ) is the distance between x and y in G . The L ( p , q , r ) -labeling number of G is the smallest number k such that G has an L ( p , q , r ) -labeling with k = max { f ( x ) : x ∈ V ( G ) } . In this paper, we obtain all the L ( p , 2 , 1 ) -labeling numbers of the infinite D -regular trees T ∞ ( D ) for p ≥ 2 and D ≥ 3 . In all cases, we also construct an optimal L ( p , 2 , 1 ) -labeling of T ∞ ( D ) .