L(d,2,1)-Labeling of Star and Sun Graphs

Abstract

For positive integer d, L(d,2,1)-labeling of a graph G is a function f from V(G) to the positive integers, f: V(G)-> {1, 2,…} such that |f(u) – f(v)| ³ d if the distance between any 2 vertices u and v is 1 (D(u,v) =1), |f(u) – f(v)| ³ 2 if  D(u,v) = 2, and |f(u) – f(v)| ³ 1 if D(u,v) = 3. The L(d,2,1)-labeling number k_d(G) of a graph G is the smallest positive integer k_d such that G has an L(d,2,1)-labeling with k_d as the maximum label. This paper presents a general k_d-value of stars K_1,n and k_d-value of sun graphs S_n for d = 3.

Keywords: L(d,2,1)-Labeling, distance, L(d,2,1)-Labeling number, sun graphs, stars.