Labeling trees with a condition at distance two

An L(h,k)L(h,k)-labeling of a graph GG is an integer labeling of vertices of GG, such that adjacent vertices have labels which differ by at least hh, and vertices at distance two have labels which differ by at least kk. The span of an L(h,k)L(h,k)-labeling is the difference between the largest and the smallest label. We investigate L(h,k)L(h,k)-labelings of trees of maximum degree ΔΔ, seeking those with small span. Given ΔΔ, hh and kk, span λλ is optimal for the class of trees of maximum degree ΔΔ, if λλ is the smallest integer such that every tree of maximum degree ΔΔ has an L(h,k)L(h,k)-labeling with span at most λλ. For all parameters Δ,h,kΔ,h,k, such that h