The L(d,1)L(d,1)-number of powers of paths

Given a graph G=(V,E)G=(V,E) and a positive integer dd, an L(d,1)L(d,1)-labelling   of GG is a function f:V→{0,1,…}f:V→{0,1,…} such that if two vertices xx and yy are adjacent, then |f(x)−f(y)|≥d|f(x)−f(y)|≥d; if they are at distance 2, then |f(x)−f(y)|≥1|f(x)−f(y)|≥1. The L(d,1)L(d,1)-number   of GG, denoted by λd,1(G)λd,1(G), is the smallest number mm such that GG has an L(d,1)L(d,1)-labelling with m=max{f(x)∣x∈V}m=max{f(x)∣x∈V}. We correct the result on the L(d,1)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)L(d,1)-labelings of graphs, Discrete Math. 220 (2000) 57–66].