on l(d,1)-labelling of trees

Given a graph g and a positive integer d, an l(d,1)-labelling of g is a function f that assigns to each vertex of g a non-negative integer such that if two vertices u and v are adjacent, then |f(u)-f(v)|>= d and if u and v are at distance two, then |f(u)-f(v)|>= 1. the l(d,1)-number of g, λd(g), is the minimum m such that there is an l(d,1)-labelling of g with f(v)⊆ {0,1,… ,m}. a tree t is of type 1 if λd(t)= δ +d-1 and is of type 2 if λd(t)>= δ+d. this paper provides sufficient conditions for λd(t)=δ+d-1 generalizing the results of wang [11] and zhai, lu, and shu [12] for l(2,1)-labelling.