L(p,2,1)L(p,2,1)-labeling of the infinite regular trees

An L(p,q,r)L(p,q,r)-labeling of a graph GG is defined as a function ff from the vertex set V(G)V(G) into the nonnegative integers such that for any two vertices x,yx,y, |f(x)−f(y)|≥p|f(x)−f(y)|≥p if d(x,y)=1d(x,y)=1, |f(x)−f(y)|≥q|f(x)−f(y)|≥q if d(x,y)=2d(x,y)=2 and |f(x)−f(y)|≥r|f(x)−f(y)|≥r if d(x,y)=3d(x,y)=3, where d(x,y)d(x,y) is the distance between xx and yy in GG. The L(p,q,r)L(p,q,r)-labeling number of GG is the smallest number kk such that GG has an L(p,q,r)L(p,q,r)-labeling with k=max{f(x):x∈V(G)}. In this paper, we obtain all the L(p,2,1)L(p,2,1)-labeling numbers of the infinite DD-regular trees T∞(D)T∞(D) for p≥2p≥2 and D≥3D≥3. In all cases, we also construct an optimal L(p,2,1)L(p,2,1)-labeling of T∞(D)T∞(D).