The -number of powers of paths

Given a graph G = ( V , E ) and a positive integer d , an L ( d , 1 ) -labelling of G is a function f : V → { 0 , 1 , … } such that if two vertices x and y are adjacent, then | f ( x ) − f ( y ) | ≥ d ; if they are at distance 2, then | f ( x ) − f ( y ) | ≥ 1 . The L ( d , 1 ) -number of G , denoted by λ d , 1 ( G ) , is the smallest number m such that G has an L ( d , 1 ) -labelling with m = max { f ( x ) ∣ x ∈ V } . We correct the result on the L ( d , 1 ) -number of powers of paths given by Chang et al. in [G.J. Chang, W.-T. Ke, D. Kuo, D.D.-F. Liu, R.K. Yeh, On L ( d , 1 ) -labelings of graphs, Discrete Math. 220 (2000) 57–66].