L(j,k)L(j,k)-labelling and maximum ordering-degrees for trees

Let GG be a graph. For two vertices uu and vv in GG, we denote d(u,v)d(u,v) the distance between uu and vv. Let j,kj,k be positive integers with j⩾kj⩾k. An L(j,k)L(j,k)-labelling   for GG is a function f:V(G)→{0,1,2,…}f:V(G)→{0,1,2,…} such that for any two vertices uu and vv, |f(u)−f(v)||f(u)−f(v)| is at least jj if d(u,v)=1d(u,v)=1; and is at least kk if d(u,v)=2d(u,v)=2. The span   of ff is the difference between the largest and the smallest numbers in f(V)f(V). The λj,kλj,k-number   for GG, denoted by λj,k(G)λj,k(G), is the minimum span over all L(j,k)L(j,k)-labellings of GG. We introduce a new parameter for a tree TT, namely, the maximum ordering-degree, denoted by M(T)M(T). Combining this new parameter and the special family of infinite trees introduced by Chang and Lu (2003) [3], we present upper and lower bounds for λj,k(T)λj,k(T) in terms of jj, kk, M(T)M(T), and Δ(T)Δ(T) (the maximum degree of TT). For a special case when j⩾Δ(T)kj⩾Δ(T)k, the upper and the lower bounds are kk apart. Moreover, we completely determine λj,k(T)λj,k(T) for trees TT with j⩾M(T)kj⩾M(T)k.