The L(2,1)L(2,1)-labeling on planar graphs

Given non-negative integers jj and kk, an L(j,k)L(j,k)-labeling of a graph GG is a function ff from the vertex set V(G)V(G) to the set of all non-negative integers such that |f(x)−f(y)|≥j|f(x)−f(y)|≥j if d(x,y)=1d(x,y)=1 and |f(x)−f(y)|≥k|f(x)−f(y)|≥k if d(x,y)=2d(x,y)=2. The L(j,k)L(j,k)-labeling number λj,kλj,k is the smallest number mm such that there is an L(j,k)L(j,k)-labeling with the largest value mm and the smallest label 0. This paper presents upper bounds on λ2,1λ2,1 and λ2,1λ2,1 of a graph GG in terms of the maximum degree of GG for several classes of planar graphs. These bounds are the same as or better than previous results for the maximum degree less than or equal to 4.