L(d,2,1)-Labeling of Simple Graphs
An L(3, 2, 1)-labeling is a simplified model for the channel assignment problem. It is a natural generalization of the widely studied L(2, 1)labeling. An L(3, 2, 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of positive integers such that for any two vertices x, y, if d(x, y) = 1, then |f(x) − f(y)| ≥ 3; if d(x, y) = 2, then |f(x) − f(y)| ≥ 2; and if d(x, y) = 3, then |f(x) − f(y)| ≥ 1. The L(3, 2, 1)-labeling number k(G) of G is the smallest positive integer k such that G has an L(3, 2, 1)-labeling with k as the maximum label. In this paper we determine the L(3, 2, 1)-labeling number for paths, cycles, caterpillars, nary trees, complete graphs and complete bipartite graphs. We also present an upper bound for k(G) in terms of the maximum degree of G.
On L(k, k-1, ..., 1) labeling of triangular lattice
Abstract An n - L ( k , k − 1 , … , 1 ) labeling of a simple graph G is a mapping f : V ( G ) → { 0 , 1 , … , n } such that | f ( u ) − f ( v ) | ≥ k + 1 − d ( u , v ) , for all u , v ∈ V ( G ) , where d ( u , v ) is the length of the shortest path connecting u and v . The L ( k , k − 1 , … , 1 ) labeling span λ k ( F ) of a family of graphs F is the minimum n for which each G ∈ F admits an n - L ( k , k − 1 , … , 1 ) labeling. For the family L 3 of all subgraphs of an infinite triangular lattice we provide upper and lower bounds of λ k ( L 3 ) for general k and show that the ratio of the upper and lower bound is at most 4 3 . The upper bound is given by providing an assignment algorithm to the vertices of the infinite triangular lattice.
On a New Subclass of p-Valent Close-to-Convex Mappings Defined by Two-Sided Inequality
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