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.
Frequency assignment problem in networks with limited spectrum
The frequency assignment problem (FAP) asks for assigning frequencies (channels) in a wireless network from the available radio spectrum to the transceivers of the network. One of the graph theoretical models of FAP is the L(3, 2, 1)-labeling of a graph, which is an abstraction of assigning integer frequencies to radio transceivers such that (i) transceivers that are one unit of distance apart receive frequencies that differ by at least three, (ii) transceivers that are two units of distance apart receive frequencies that differ by at least two, and (iii) transceivers that are three units of distance apart receive frequencies that differ by at least one. The relaxation of the L(3, 2, 1)-labeling called the (s, t, r)-relaxed k-L(3, 2, 1)-labeling is proposed in this paper. This concept is a generalization of the (s, t)-relaxed k-L(2, 1)-labeling (Lin in J Comb Optim 2016, doi:10.1007/s10878-014-9746-9). Basic properties of (s, t, r)-relaxed k-L(3, 2, 1)-labeling are discussed and optimal (s, t, r)-relaxed k-L(3, 2, 1)-labelings for paths and some cycles as well as for the hexagonal lattice and the square lattice are determined.
Kidney and urinary tract diseases in the newborn
Kidney and urinary tract diseases in the newborn / , Kidney and urinary tract diseases in the newborn / , کتابخانه دیجیتال جندی شاپور اهواز