The L (2, 1)-labeling of K_{1, n}-free graphs and its applications

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