Circular Distance-Two Labelling of Book Graphs Related to Code Assignment in Computer Wireless Networks

Abstract

Let d be a positive real number. An L(1,d)-labeling of a graph G is an assignment of nonnegative real numbers to the vertices of G such that the adjacent vertices are assigned two different numbers (labels) whose difference is at least one, and the difference between numbers (labels) for any two distance-two vertices is at least d. The minimum range of labels over all L(1,d)-labelings of a graph G is called the L(1,d)-labeling number of G, denoted by λ_(1,d)(G). The L(1,d)-labeling with d≥1 of graph arose from the code assignment problem of computer wireless network and the L(1,d)-labeling with 0<d≤1 of graph is motivated by channel assignment problem. Let σ and d be two positive real numbers, a circular σ-L(1,d)-labeling of a graph G is a function f: V(G)→[0,σ) satisfying |f(u)-f(v)|σ≥1 if u v∈E(G), and |f(u)-f(v)|σ≥d if u and v are distance two apart, where |f(u)-f(v)|σ=min{|f(u)-f(v)|, σ-|f(u)-f(v)|}. The circular L(1,d)-labeling number of a graph G, denoted by σ_(1,d) (G), is the minimum σ such that there exists a circular σ-L(1,d)-labeling of G. In this paper, the code assignment of 3-D computer wireless network is abstracted as the circular L(1,d)-labeling of book graph, and the authors determined the circular L(1,d)-labeling numbers of book graph for any positive real number d≥2 basing on the properties and constructions of book graphs.