Lp,q-Label Coloring Problem with Application to Channel Allocation

Zhenbin Liu,Yuqiang Wu

doi:10.1155/2020/5369859

Abstract

In this paper, the Lp,q-coloring problem of the graph is studied with application to channel allocation of the wireless network. First, by introducing two new logical operators, some necessary and sufficient conditions for solving the Lp,q-coloring problem are given. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Second, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Finally, the obtained result is applied to analyze channel allocation of the wireless network. Furthermore, an illustration example is given to show the effectiveness of the obtained results in this paper.

Highlights

It is well known that the coloring problem is a basic and classical problem in graph theory
Graph coloring is originated from famous conjecture called four-colour conjecture [1] and widely used in many real-life areas [2,3,4], such as scheduling and timetabling in engineering, air traffic flow management, and channel allocation of mobiles. ere are various forms of graph coloring, such as set coloring, list coloring, T-coloring, and L(n1, n2, · · ·, ns)-coloring. e labeling problems of graphs arise in many networking and telecommunication contexts. e channel allocation problem is first formulated as a graph coloring problem by Hale [5]
It is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. en, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form

Summary

Introduction

It is well known that the coloring problem is a basic and classical problem in graph theory. Cheng et al and Li et al provided a new mathematical method, which is called the semitensor product with matrices [11,12,13] to study logical systems [14,15,16,17,18,19,20,21,22,23], probability logical networks [24, 25], game theory [26, 27], coloring problem [1, 10, 28], and some other related fields [29,30,31]. Is paper studies the L(p, q)-coloring problem of the graph with application to channel allocation of the wireless network.

Preliminaries

Main Results

Illustrative Example