Optimizing distance constraints frequency assignment with relaxation

In a typical wireless telecommunication network, a large number of communication links is established with a limited number of available frequencies. The problem that addresses assigning available frequencies to transmitters such that interference is avoided as far as possible is called the frequency assignment problem. The problem is usually modeled as a graph coloring (labeling) problem. We study in this paper the $(s,t)$-relaxed $L(2,1)$-labeling of a graph which considers the situation where transceivers that are very close receive frequencies that differ by at least two while transceivers that are close receive frequencies that differ by at least one. In addition, the model allows at most $s$ (resp. $t$) anomalies at distance one (resp. two). The objective of the model is to minimize the span of frequencies in a corresponding network. We show that it is NP-complete to decide whether the minimal span of a $(1,0)$-relaxed $L(2,1)$-labeling of a graph is at most $k$. We also prove that the minimal span of this labeling for two classes of graphs is bounded above with the the square of the largest degree in the graph of interest. These results confirm Conjecture 6 and partially confirm Conjecture 3 stated in [W. Lin, On $(s, t)$-relaxed $L(2, 1)$-labeling of graphs, J. Comb. Optim. 2016 (31) 1--22].

A receptance-based method for frequency assignment via coupling of subsystems

This paper presents a theoretical study of the frequency assignment problem of a coupled system via structural modification of one of its subsystems. It deals with the issue in which the available modifications are not simple; for example, they are not point masses, grounded springs, or spring-mass oscillators. The proposed technique is derived based on receptance coupling technique and formulated as an optimization problem. Only a few receptances at the connection ends of each subsystem are required in the structural modification process. The applicability of the technique is demonstrated on a simulated rotor system. The results show that both bending natural frequencies and torsional natural frequencies can be assigned using a modifiable joint, either separately or simultaneously. In addition, an extension is made on a previously proposed torsional receptance measurement technique to estimate the rotational receptance in bending. Numerical simulations suggest that the extended technique is able to produce accurate estimations and thus is appropriate for this frequency assignment problem of concern.

Efficient Algorithm For L(3,2,1)-Labeling of Cartesian Product Between Some Graphs

L(h,k) Labeling in graph came into existence as a solution to frequency assignment problem. To reduce interference a frequency in the form of non negative integers is assigned to each radio or TV transmitters located at various places. After L(h,k) labeling, L(h,k, j) labeling is introduced to reduce noise in the communication network. We investigated the graph obtained by Cartesian Product betweenCompleteBipartiteGraphwithPathandCycle,i. e.,Km,n×Pr andKm,n×Cr byapplying L(3,2,1)Labeling. The L(3,2,1) Labeling of a graph G is the difference between the highest and the lowest labels used in L(3,2,1) and is denoted by λ3,2,1(G) In this paper we have designed three suitable algorithms to label the graphs Km,n × Pr and Km,n × Cr . We have also analyzed the time complexity of each algorithm with illustration.