L(3,2,1) Labeling of Firecracker Graph

Abstract

Let G = (V, E) be a graph. An L(3,2,1) labeling of G is a function f : V → N ∪ {0} such that for every u, v ∈ V , |f(u) − f(v)| ≥ 3 if d(u, v) = 1, |f(u) − f(v)| ≥ 2 if d(u, v) = 2, and |f(u) − f(v)| ≥ 1 if d(u, v) = 3. Let k ∈ N, a k − L(3, 2, 1) labeling is a labeling L(3,2,1) where all labels are not greater than k. An L(3,2,1) number of G, denoted by λ(3,2,1)(G), is the smallest non-negative integer k such that G has a k − L(3,2,1) labeling. In this paper, we determine λ(3,2,1) of firecracker graphs.