The L(2,1)L(2,1)-labeling of K1,nK1,n-free graphs and its applications

An L(2,1)L(2,1)-labeling of a graph GG is a function ff from the vertex set V(G)V(G) into the set of nonnegative integers such that |f(x)−f(y)|≥2|f(x)−f(y)|≥2 if d(x,y)=1d(x,y)=1 and |f(x)−f(y)|≥1|f(x)−f(y)|≥1 if d(x,y)=2d(x,y)=2, where d(x,y)d(x,y) denotes the distance between xx and yy in GG. The L(2,1)L(2,1)-labeling number, λ(G)λ(G), of GG is the minimum kk where GG has an L(2,1)L(2,1)-labeling ff with kk being the absolute difference between the largest and smallest image points of ff. In this work, we will study the L(2,1)L(2,1)-labeling on K1,nK1,n-free graphs where n≥3n≥3 and apply the result to unit sphere graphs which are of particular interest in the channel assignment problem.