A thesis submitted

to Kent State University in partial

fulfillment of the requirements for the

degree of Master of Science

By

Uday Kiran Viyyure

May 2008

Thesis written by

Uday Kiran Viyyure

M.S., Kent State University, USA, 2008

Approved by

Dr. Feodor F. Dragan , Advisor

Dr. Robert A. Walker , Chair, Department of Computer Science

Dr. Jerry Feezel , Dean, College of Arts and Sciences

ii

LIST OF FIGURES ...... iv

LIST OF TABLES ...... vi

ACKNOWLEDGEMENTS ...... vii

CHAPTER 1 INTRODUCTION...... 1

CHAPTER 2 PRELIMINARIES ...... 5

2.1 Distance K-Chromatic Number Problem ...... 7

2.2 The L(h,k) Graph Coloring Problems...... 8

CHAPTER 3 COLORING METHODS ...... 10

3.1 The L(1) Graph Coloring ...... 10

3.2 The L(1,1) Graph Coloring ...... 12

3.3 The L(1,1,1) Graph Coloring ...... 16

3.4 The L(2,1) Graph Coloring ...... 19

3.5 The L(2,1,1) Graph Coloring ...... 22

3.6 The L(3,1,1) Graph Coloring ...... 28

CHAPTER 4 CONCLUSIONS AND FUTURE RESEARCH ...... 36

APPENDIX A FREQUENCY ASSIGNMENT TOOL ...... 37

REFERENCES...... 40

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LIST OF FIGURES

Figure1. Cells showing co-channel and adjacent channel interference…………………..3

Figure 2. Hexagonal cell structure of a …………………………………4

Figure 3. Cellular graph network corresponding to a cellular network of Figure 2……..5

Figure 4. Clique for L(1) coloring...... 11

Figure 5. L(1) coloring for a 5×5 cellular network...... 11

Figure 6. L(1) coloring for a 7×7 cellular network...... 12

Figure 7. L(1) coloring for a 10×10 cellular network...... 12

Figure 8. Clique for the L(1,1) coloring of a cellular graph network...... 13

Figure 9. Color Spectrum for the L(1,1) coloring...... 14

Figure 10. L(1,1) coloring of a cellular graph network...... 14

Figure 11. L(1,1) coloring of a 5×5 cellular network...... 15

Figure 12. L(1,1) coloring of a 7×7 cellular network...... 15

Figure 13. L(1,1) coloring of a 10×10 cellular network...... 15

Figure 14. Clique for L(1,1,1) coloring in a cellular graph network...... 16

Figure 15. Honey comb cellular graph with L(1,1,1) graph coloring technique...... 17

Figure 16. Color spectrum used for L(1,1,1) coloring...... 18

Figure 17. A 5×5 cellular network with L(1,1,1) coloring...... 18

Figure 18. A 7×7 cellular network with L(1,1,1) coloring...... 18

Figure 19. A 10×10 cellular network with L(1,1,1) coloring...... 19

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Figure 20. Color spectrum used in L(2,1) Coloring...... 20

Figure 21. A 5×5 cellular network with L(2,1) coloring...... 20

Figure 22. A 7×7 cellular network with L(2,1) coloring...... 21

Figure 23. A 10×10 cellular network with L(2,1) coloring...... 21

Figure 24. Clique of a cellular graph with the integers of the colors associated...... 22

Figure 25. Color spectrum used for L(2,1,1) graph coloring...... 23

Figure 26. Cliques of a cellular graph with L(2,1,1) coloring...... 23

Figure 27. A 5×5 cellular network with L(2,1,1) coloring...... 24

Figure 28. A 7×7 cellular network with L(2,1,1) coloring...... 24

Figure 29. A 10×10 cellular network with L(2,1,1) coloring...... 25

Figure 30. Coloring spectrum for L(3,1,1) coloring...... 29

Figure 31. A 5×5 cellular network with L(3,1,1) coloring...... 29

Figure 32. A 7×7 cellular network with L(3,1,1) coloring...... 29

Figure 33. A 10×10 cellular network with L(3,1,1) coloring...... 30

Figure 34. Tool interface with number of rows and columns entered...... 34

Figure 35. Tool generating graph without coloring...... 37

Figure 36. Tool coloring the graph with L(3,1,1) coloring method...... 37

Figure 37. Tool coloring the graph with L(3,1,1) coloring method...... 38

v

LIST OF TABLES

Table 1. Relationship between Manhattan distance and Minimum distance...... 26

Table 2. Relationship between consecutive vertices...... 26

Table 3. Manhattan distance between consecutive colors for L(3,1,1) coloring...... 31

Table 4. Manhattan distance between colors (a,b), where |a-b| = 2...... 32

vi

Acknowledgement

I would like to take this opportunity to express my sincere gratitude to my advisor Dr.

Feodor F. Dragan for kindly providing guidance throughout my research work. I thank him for his continuous encouragement and making this thesis possible. It was a great pleasure to work on my thesis under his supervision.

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CHAPTER 1

Introduction

Technological advances and the tremendous growth of network and

wireless terminals have caused the wireless communications and mobile computing to

grow rapidly. The present trends in the industry to provide

information access from any where to any place in the world and shifting of present

applications to a multimedia environment has caused a rapid growth in the mobile users

community. This enormous growth in filed has put a lot of constraints

on the availability of the spectrum. Due this the frequency reuse and

channel assignment problems and algorithms have been studied extensively in the last

two decades by radio and electrical engineers, operations researchers, graph theorists and

computer scientists. Different people approached this problem in different ways

(1) Classic methods like graph coloring and integer programming.

(2) Heuristic methods like neural networks genetic algorithms.

(3) Local search such as simulated annealing and Tabu search.

(4) Constrained programming.

1

2

In cellular networks a geographical area is divided into smaller service areas called cells and each of these cell’s have a base station and all the wireless terminals or the users in those cells communicate with their corresponding cell area base stations. For these communication links to be established the available frequency spectrum should be used and reused very efficiently. The efficient reuse in the spectrum helps to reduce the cost of service by reducing the number of base stations and also accommodating more number of users per base stations.

A channel assignment problem or the frequency assignment problem is nothing but the task of assigning frequency from a radio spectrum to a set of transmitters and receivers satisfying certain conditions. In order to divide a radio spectrum in an optimal way many techniques have been given in literature, such as frequency division, time division and code division. In Frequency division the spectrum is divided in to disjoint frequency bands. In time division the same channel is used by different base stations at different times and thus attaining channel allocation with respect to time. In channel division the channel allocation is achieved by using different modulation code. We can also achieve channel assignment by a combination of above techniques. Here in this paper we discuss with respect to frequency division, so we view the channel assignment as nothing but frequency assignment problem.

Every cell’s base station is equipped with a low power transmitter which makes the reuse of frequency possible but there are some factors which affect the reusability.

3

The prime factors which effect frequency assignment and channel reusability are Co-

Channel interference and adjacent channel interference. All the channels which use the same frequency are called co-channels. The minimum distance at which two base stations can co-use same channel with acceptable interference is called the “Co-Channel Reuse

Distance” σ. We can reuse the channel in this way because of propagation path-loss in the radio environment.

3 4 F1

6

2 F2 F1 1 5

Fig 1: Cells showing co-channel and adjacent channel interference.

In the above diagram we can see that station 2 and 3 use the same channel and station 5

uses a different channel. Here 2 and 3 are co-channels and since they are considerable

apart there would be no problem but if station 5 uses the F1 frequency then station 1

might get effected by the channels from station 2 and station 5 and this will cause co-

channel interference. For this reason the channels should be “ σ” apart. Adjacent channel

Interference is due to the signal energy from an adjacent channel spilling over into the

channel. In our discussion we consider only co-channel interference for channel

assignment.

4

Here we consider channel assignment for general static cellular networks. These are considered as honeycomb networks and with respect to graphs they are represented as hexagonal grids and the center of these grids are the base stations to which the channel assignment should be done. Usually cellular networks are modeled as infinite triangular lattice and the reason for this that cells are usually distributed uniformly in geographic area. The hexagonal shape is taken into consideration for the shape of cell because the attenuation of the radio signals occurs in a circular manner and so a base station reachability area can be best depicted as a hexagon. The triangular lattice is nothing but the planar dual of the resulting voronoi diagram.

Fig 2: Hexagonal cell structure of a cellular network

5

Fig 3: Cellular graph Network corresponding to Cellular network of Fig 2.

CHAPTER 2

Preliminaries

In the graph coloring approach towards channel assignment problem, we consider an undirected G =(V, E) where V represent the set of vertices in graph which are the base station and E represent the set of edges which correspond to pairs of stations whose transmission regions intersect. In this graph the channels are represented by colors, which are assigned non negative numbers and usually the smallest color is 0. The two problems of channel assignment i.e. co-channel interference and adjacent channel interference is tackled by taking following approach. The co-channel interference is overcome by separating the channels by a distance “σ” called co-channel reuse distance and the interference phenomena problem is approached by imposing channel separation.

The gap between the two channels is usually inversely proportional to the distance between the two stations. Usually one unit distance in graph coloring problem is measured as distance between two nodes. Since we consider the static graph coloring method where the cells are fixed and are equidistant from all its neighboring nodes the distance is considered as constant. Let us consider two vertices u and v, let f(u) and f(v) be the frequencies assigned to them and i be the distance between them and i < σ, then the minimum channel separation ( δi ) should satisfy the following condition |f(u)-f(v)| ≥

6 7

δi. Channel assignment should be in such a way that satisfies both co-channel re-use and adjacent channel interference conditions and difference between the highest and the lowest frequencies assigned should be kept as low as possible.

For a given graph G = (V, E) let d (u, v) be the distance between vertices u and v, where the d is given as the number of edges on a shortest path between u and v. Given σ and the channel separation vector (δ1, δ2, .., δ σ-1) of non negative integers, an L ( δ1, δ2 ,…,

δ σ-1) coloring of the graph G is a function f from the vertex set V to the set Λ={0,…., λ} of colors, such that |f(u)-f(v)| ≥ δi if d(u,v) = i for i=0,1,2…. σ-1. Alan et al [8] have defined the channel assignment problem as it is the problem of finding an optimal L ( δ1, δ2,…, δ

σ-1) coloring of G, that is one which minimizes the largest color λ. Note that since Λ contains 0, the number of colors used is λ+1, but not all the colors are used to color the

graph. The channel assignment problem is shown to be NP-complete because this

problem was reduced to the graph K-coloring problem [8] and therefore it is

computationally intractable. We review L(1), L(1,1), L(1,1,1), L(2,1), L(2,1,1)-coloring

know from literature and based on this we developed a technique for L(3,1,1) coloring

which is, to the best of our knowledge, new. The problem which we discuss can also be

modeled as a Distance K-chromatic number problem [8].

2.1 Distance K-Chromatic Number Problem

Given a graph G = (V, E) and an integer K, the Distance K chromatic number of

the graph is the fewest number of colors needed to color the nodes of the graph so that no

8

two nodes of the graph have the same color if the shortest path length between the nodes is less than or equal to k. Such a coloring of the nodes of the graph is known as a proper coloring. How we can interpret this problem is if 2 base stations are more than K distance unit separate, then they can use the same color. The frequency separation between adjacent cells varies inversely to the distance between them. It is known [8] for distance

K color, a cellular graph can be colored with ¾(K+1) 2 +1/4 colors if K is even and

¾(K+1) 2 colors if K is odd. This is also proven by Alan et.al [8].

The Channel assignment problem has been approached in many different ways but the common was reducing the span of colors. Some researchers instead of reducing the span tried to minimize the order i.e. using minimum number of colors. Usually minimizing the span means not using all the colors in the given range of colors [1].

Minimizing the order means in the given range of colors to be used we need to use all the colors and in span coloring, color we miss is usually called hole in the span [2]. The other approaches are consider the color set as a cyclic interval, i.e. the distance between two labels i,j belong to the set {0,1,…. σ } defined as mini {|i-j| - |i-j|) [3], use a more general

model in which the coloring numbers and the distance between the nodes are real

numbers [4]. In some methods some nodes are already colored and then they try to see

whether this can be used to color the entire graph. In some papers they approach the

problem by studying another parameter called edge-span, defined as the minimum over

all feasible labeling of the max {|f(u)-f(v)|: (u,v) Є E(G)} [5]. In some papers the problem

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is approached by considering that all the colors should be used equally (i.e. equitable coloring) [6].

2.2 The L(h,k) Graph Coloring Problems

The classic graph coloring problem is the one where condition on the coloring of the is placed only on the adjacent nodes. In wider range of problems and for more generalization of graph coloring problem for solving wider range of problems more restrictions are placed i.e. conditions are placed not only on the adjacent nodes but also on the nodes that are at a distance of 2 apart. This problem is also called with different names in different papers. They are: L(h, k) coloring problem, distance-2 coloring problem and D2- vertex coloring problem (when h=k=1), radio coloring problem and λ- coloring problem (when h=2 and k=1). The definition of this problem is given in [7] as follows.

Definition :

Given a graph G = (V, E) and two nonnegative integers h and k, an L (h, k) -

labeling is an assignment of nonnegative integers to the nodes of G such that adjacent

nodes are labeled using colors at least h apart and vertices at distance 2 are labeled using

colors at least k apart. The aim of the L (h,k) - labeling problem is to minimize the span σ h,k (G), i.e. the difference between the largest and the smallest used colors. The minimum span over all possible labeling functions is denoted by λ h,k (G) and is called λ h,k number of G.

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Under this class of problems we study the L(1) , L(1,1) and L(2,1) coloring problems. The L(1), L(1,1) and L(1,1,1) channel assignment problems we discuss come under the class L(1,…1 k) coloring problem. In [8] it is given that lower bound for

L(1,…1 k) problem is also lower bound for L( δ,…1 k) and from this [9] stated that the graph we get by joining all the nodes which are at a distance of σ-1 is considered as a

clique and the size i.e. the number of vertices of that clique will give the lower bound on

the number of colors required. In the coloring problems we discuss below this approach

is taken, i.e. we find a clique and try to color the entire graph with the help of this clique.

CHAPTER 3

Coloring Methods

3.1. The L(1) Graph coloring

This is the classic graph coloring problem. The aim of this coloring is to color the graph such that no two adjacent nodes or vertices have the same color. The maximum number of colors required to color cellular graph under this condition is 3. Since the cellular graph representing a clique is a three vertices triangular lattice we can say that to color the three vertices we need at least three colors and this is pretty straight forward.

Here we take this as a sub-graph and color the entire cellular grid. This coloring when seen as L(h,k) coloring method we take h=1 and k=0 and what this means is that the adjacent nodes should be at least 1 color distance apart and there is no coloring restriction on any other nodes. In figure 4 below the clique for the L(1) is given and in Fig 5 a 5 ×5 graph which represents a cellular network with circles representing the base stations and the edges represents the link between two base station which is uniform. The Fig: 6 and

Fig: 7 represent the colored cellular network graph of size 7 ×7 and 10 ×10 size. Here we approached the coloring of the graph by coloring the (0, 0) node with one of the colors

11 12

and the next set of nodes i.e. the diagonal nodes (1, 0) and (0, 1) are colored with second color. The next set of nodes i.e. (2, 0) (1, 1) and (0, 2) are colored with the third color.

The coloring of the nodes is repeated with one color representing a diagonal and the next diagonal representing the other color and the previous diagonal representing the third color, and the coloring repeats itself until all the nodes are colored. The colors which we used here are white, green and red.

Fig: 4 Clique for L (1) Coloring.

Fig 5: L (1) Coloring for a 5 ×5 Cellular Network.

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Fig 6: L (1) Coloring for a 7 ×7 Cellular Network.

Fig 7: L (1) Coloring for a 10 ×10 Cellular Network.

3.2 The L(1,1) Graph Coloring

Here in this coloring method the condition which should be satisfied is that the adjacent nodes should be separated at least by one unit and also those which are at a

14

distance 2 should also be separated at least by one unit. For this type of coloring the minimum number of colors required is 7. The problem to minimize the order of colors for this type of coloring method is proved to be NP-complete for general graphs since it was reduced to 3-SAT problem [10]. The number of colors used for this condition, for cellular graph networks is 7. The coloring of cellular network for this condition is approached by Sen. et al [11] by finding the tile or the clique for this coloring condition which is shown in the Fig:8 and then coloring the tile and repeating the tile coloring for whole cellular network.

Fig 8: The clique for the L (1, 1) coloring for cellular graph network.

The colors used for this condition are from 0 to 6. The coloring is done by assigning 0 to the top left node and next node which is in the same row is assigned 1. The next color 3 is assigned to the next row first node. In this order all the nodes in the clique are colored.

Now the entire cellular graph network is divided into these cliques. After the graph network is divided into cliques each clique is colored individually and by doing so we

15

color the entire graph. The colors of the two adjacent nodes does not conflict or violate the condition of the L (1, 1) color. In figure 11, 12 and 13 the cellular graphs are colored with the L (1, 1) coloring condition. The Colors used for this coloring are given below

Fig 9: Colors used for the L (1,1) coloring

Fig 10: L (1,1) coloring of a cellular graph network .

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Fig 11: L (1,1) Coloring of a 5×5 cellular graph.

Fig 12: L (1, 1) coloring of a 7×7 cellular graph.

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Fig 13: L (1, 1) coloring of a 10×10 cellular graph network.

3.3 The L (1,1,1) Graph coloring

In the L (1, 1, 1) graph coloring the condition that coloring should satisfy is that the adjacent nodes should be of different colors and the nodes at distance 2 and 3 should be also of different colors. In this coloring method the reuse distance is 4 (i.e. σ = 4). The clique for this kind of coloring found out by Sen. et al [11] and it is given in the figure 14.

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Fig 14: The Clique for L(1,1,1) coloring in a Cellular graph network.

In [11] they give the lower bound for the coloring schema as 12. The above clique uses colors from 0 to 11 which is nothing but minimum order of colors. The coloring method followed is same as that of the L(1,1) coloring. The nodes of the first row are colored from left to right in the increasing order and in this order next three rows are colored. The below graph gives a clear picture of the coloring technique we discussed, were given graph is divided into cliques and these cliques are colored which in turn colors the entire graph and the condition of the L(1,1,1) coloring are not violated i.e. the reuse distance is maintained to be 4.

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Fig 15: A honey comb Cellular graph with L(1,1,1) graph coloring technique.

Using this approach we colored various size cellular graphs and the results are given in the diagrams given below. Figure 17 represents a 5 ×5 cellular graph network colored using L (1, 1, 1) coloring. Figure 18 and 19 represent 7 ×7 and 10 ×10 size cellular graph networks colored using L(1,1,1) condition. The Colors used for the L(1,1,1) coloring are as given below. The colors are from 0 to 11 and so the total number of colors used here is

12.

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Fig 16: Colors used for L(1,1,1) Coloring.

Fig 17: A 5 ×5 Cellular graph with L(1,1,1) Coloring

Fig 18: A 7×7 Cellular graph with L(1,1,1) Coloring.

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Fig 19: A 10 × 10 Cellular graph Networks Colored in L(1,1,1) coloring.

3.4 The L(2,1) Graph Coloring

The L(2,1) coloring has been very extensively studied in literature by many and

has been addressed in different papers under different names like radio coloring, λ coloring . This problem was first brought in to light by J.R.Griggs and R.K.Yeh [12] after that the problem was approached by many for different kinds of graphs. It was proven in [13] that it is NP complete to color a graph with L(2,1) condition. The L(2,1) condition is where the nodes of the graph which are adjacent should at least be at a distance of 2 units in the color spectrum. The reuse distance of this graph is 3 (i.e. σ=3).

This problem can be defined as follows[14]:

Definition :

An L(2,1) labeling of a graph G is a function ƒ from the vertex set V(G) to the set of all nonnegative integers such that | ƒ(u)- ƒ(v)|≥1 if d(u, v)=2 and | ƒ(u)- ƒ(v) | ≥ 2

22

if d(u, v) = 1.

Alan et al [8] approached this problem by extending their concept of coloring a bidimensional grids in L(2,1) coloring to the cellular graph networks and they gave an algorithm for coloring a cellular grid network of “r” rows and “c” columns. The total number of colors used for this coloring technique is 9.

Algorithm for coloring cellular graph network with L(2,1) coloring :

• If r ≥ 4 and c ≥ 4, or r = 3 and c ≥ 5, or r ≥ 5 and c = 3 then assign to each vertex u = (i , j) the color ƒ(u) = (3i+2j) mod 9.

The figures 21, 22 and 23 below illustrates the L(2,1) coloring on cellular networks of

different sizes and the spectrum of colors used in this coloring is given in the Fig 20.

Fig 20: Spectrum of colors used in the L(2,1) coloring.

Fig 21: A cellular grid of Size 5×5 colored by L(2,1) coloring algorithm.

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Fig 22: A cellular grid of size 7 ×7 colored by L(2,1) coloring algorithm.

Fig 23: A cellular grid of size 10 ×10 colored by L(2,1) coloring algorithm.

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3.5 The L(2,1,1) Coloring of cellular graph network

The L(2,1,1) coloring of a graph is done in such a way that the nodes which are

adjacent should be separated at least by 2 units in the color frequency spectrum. The

nodes/ base stations/vertices which are at distance 2 should be having different colors and

also the nodes which are at distance 3 units should also have different colors. The reuse

distance of the colors in this coloring schema is 4 i.e. σ = 4. The coloring solution for this

graph was given by Alan et al [8]. They approached the problem by first finding the

clique and then coloring the clique by a suitable algorithm. The coloring technique of the

clique can be extended to any size cellular graph. The Fig: 24 below gives the clique for

L (2, 1, 1) coloring of a cellular grid. The coloring spectrum for this graph is given

below.

Fig 24: Clique of a cellular graph with the integers of the colors associated.

The color spectrum used for this graph is given below which consists of 12 colors.

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Fig 25: Color spectrum used for L(2,1,1) graph coloring.

Fig 26: Clique of a cellular Graph with L(2,1,1) coloring.

The above coloring requires 12 colors and in the coloring spectrum they color from 0 to

11 with no holes i.e. it is the minimized order for the L(2,1,1) coloring. In the above Fig

24 we can observe that two adjacent nodes have a difference of more than one in the coloring integer unit. The one algorithm that was given by Alan et al for coloring any given cellular graph is given below.

Algorithm for L(2,1,1) coloring of a cellular graph:

If r ≥ 4 and c ≥ 4 then assign to each vertex u = (i, j) the color

ƒ (u) = 0 if (i + j) ≡ 2 mod 6, i is even, and j is even. ƒ (u) = 1 if (i + j) ≡ 0 mod 6, i is even, and j is even. ƒ (u) = 2 if (i + j) ≡ 4 mod 6, i is even, and j is even. ƒ (u) = 3 if (i + j) ≡ 1 mod 6, i is odd, and j is even. ƒ (u) = 4 if (i + j) ≡ 3 mod 6, i is odd, and j is even. ƒ (u) = 5 if (i + j) ≡ 5 mod 6, i is odd, and j is even. ƒ (u) = 6 if (i + j) ≡ 5 mod 6, i is even, and j is odd. ƒ (u) = 7 if (i + j) ≡ 2 mod 6, i is odd, and j is odd. ƒ (u) = 8 if (i + j) ≡ 4 mod 6, i is odd, and j is odd.

26

ƒ (u) = 9 if (i + j) ≡ 1 mod 6, i is even, and j is odd. ƒ (u) = 10 if (i + j) ≡ 3 mod 6, i is even, and j is odd. ƒ (u) = 11 if (i + j) ≡ 0 mod 6, i is odd, and j is odd.

The figures show us the colored graph of different sizes obtained by applying the above algorithm.

Fig 27: A 5 ×5 Cellular graph network colored by L(2,1,1) Coloring method.

Fig 28: The L(2,1,1) coloring of a 7 ×7 cellular graph.

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Fig 29: The L(2,1,1) coloring of a 10 ×10 cellular graph network.

From the above discussion we can see that tiling method has been applied to color the graphs. If we carefully observe the graphs we can see that each vertex has six adjacent vertices. We can also observe that each adjacent vertex is separated by a color difference of at least 2 or more. In order to prove that the above given algorithm for L (2,1,1) works for any given cellular graph we have to prove that

(a) The distance between a pair of vertices with consecutive colors should be at

least 2.

(b) The distance between two vertices colored by the same color should be greater

than 3.

To prove the above two points we use the Manhattan distance concept. Manhattan distance m (u, v) is defined as the shortest distance between u and v vertices, where the distance is calculated by using only horizontal and vertical edges but not the diagonal edges. Therefore by observation we can state the relationship between the Manhattan

28

distance and minimum distance in the form of a table which is given below. Here the minimum distance is the length of a shortest path between two vertices where in this path the diagonal edges can also be included.

Manhattan distance Minimum Distance

1 1 2 >=1 3 >=2 4 >=2 5 >=3 6 >=3 Table 1: Relationship between Manhattan distance and Minimum distance.

In order to prove that all consecutive colors are separated by a distance 2, we need

to check the minimum distance between consecutive colors. By examining the L(2,1,1)

algorithm function f(u) we have the following table below.

Consecutive colors Manhattan Distance i Parity j Parity

(0,1) >=2 Both even Both even (1,2) >=4 Both even Both even (2,3) >=3 Even/odd Both even (3,4) >=2 Both odd Both even (4,5) >=2 Both odd Both even (5,6) ? Even/odd Even/odd (6,7) >=3 Even/odd Both odd (7,8) >=2 Both odd Both odd (8,9) >=3 Odd/even Both odd (9,10) >=2 Both even Both odd (10,11) >=3 Even/odd odd Table 2: Relationship between consecutive vertices.

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We can notice the special case of color 5 and color 6, both are 5 mod 6. It is possible that their Minimum distance is 0, but if it is 0 they should demonstrate same parity properties for i or j which they don’t. Is it possible that their minimum distance is 1? No. If their minimum distance is 1, they won’t have the same value for mod 6. Therefore, their minimum distance should at least be 2.

For the pairs (1,2) (2,3) (5,6) (6,7) (8,9) and (10,11) each has Manhattan distance no less than 3. According to table 1, their minimum distance should be at least two. For the remaining consecutive pair of colors (0,1), (3,4), (4,5), (7,8) and (9,10) each has

Manhattan distance no less than 2, but each has the same parity property for i or j coordinates of its two colors. By observing Fig 28, we can say that for any two vertices

(i,j) and (i’,j’) they are 2. Manhattan distance away their distance cannot be 1 unless i and i’ (j and j’) have different parities. Thus we can say that even these pairs are at least 2 units distance apart. Thus we can say that the distance between a pair of vertices with consecutive colors is at least 2.

In addition the Manhattan distance between two vertices of the same color is greater than or equal to 6. From f(u) of the algorithm we also know that all vertices of the same color have the same parity property for i or j co-ordinate. Therefore from these observations we can say that the vertices with the same color are separates by more than

3 units of distance. Thus proving that the second statement is also true. Therefore function f(u) for the L(2,1,1) coloring holds true for any given cellular graph.

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In the next coloring scheme which is L(3,1,1) we use the same tiling method and also we prove that the function f(u) given by us in algorithm is correct by using the same methodology which we used to prove that the function f(u) for L(2,1,1) is correct.

3.6 The L(3,1,1) Graph coloring

In this graph coloring method the condition to be satisfied by the coloring is that the integers assigned to the adjacent nodes should at least have difference 3 or more and the reuse distance in this method is 4 units (i.e. σ = 4). The nodes which are at distance 2

and 3 apart should be of different colors. For this coloring we did an exhaustive search

and found out that the span of the color spectrum stretches from 0 to 13. When we use a

spectrum from 0 to 12 the coloring fails for a 6 ×6 cellular grid. In this coloring method we don’t use the entire set of colors, the colors 4, and 9 are not used and are considered holes in the span which is from 0 to 13. The coloring function ƒ for L(3,1,1) coloring is given below.

Algorithm for L(3,1,1) coloring:

ƒ (u) = 10 if (i + j) ≡ 2 mod 6, i is even, and j is even. ƒ (u) = 0 if (i + j) ≡ 0 mod 6, i is even, and j is even. ƒ (u) = 8 if (i + j) ≡ 4 mod 6, i is even, and j is even. ƒ (u) = 6 if (i + j) ≡ 1 mod 6, i is odd, and j is even. ƒ (u) = 1 if (i + j) ≡ 3 mod 6, i is odd, and j is even. ƒ (u) = 13 if (i + j) ≡ 5 mod 6, i is odd, and j is even. ƒ (u) = 12 if (i + j) ≡ 5 mod 6, i is even, and j is odd. ƒ (u) = 11 if (i + j) ≡ 2 mod 6, i is odd, and j is odd. ƒ (u) = 7 if (i + j) ≡ 4 mod 6, i is odd, and j is odd. ƒ (u) = 5 if (i + j) ≡ 1 mod 6, i is even, and j is odd. ƒ (u) = 3 if (i + j) ≡ 3 mod 6, i is even, and j is odd.

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ƒ (u) = 2 if (i + j) ≡ 0 mod 6, i is odd, and j is odd.

The color spectrum used for this coloring method problem is given below in Fig 29. In

Fig 30, 31 and 32 we give cellular graphs with L(3,1,1) coloring.

Fig 30: Coloring spectrum for L(3,1,1) coloring.

Fig 31: The L(3,1,1) coloring of a 5 ×5 cellular graph network.

Fig 32: The L(3,1,1) coloring of a 7 ×7 cellular graph network.

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Fig 33: The L(3,1,1) coloring of a 10 ×10 cellular graph network.

In order to prove that the above coloring technique is correct we need to prove that following three statements are true.

(a) Distance between a pair of vertices with consecutive colors should be at least 2.

(b) Distance between a pair of vertices with colors (a,b) where |a-b|=2, should be at

least 2.

(c) Distance between the vertices that have the same color should be greater than 3.

The approach we take to prove the correctness of this coloring method is same as that we have taken for L(2,1,1) coloring since both take the same tiling approach to color the graph vertices.

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First, the Manhattan distances between consecutive colors are listed in the following

table:

Consecutive Colors Manhattan i Parity j Parity Distance (0,1) >=3 Even/odd Both even (1,2) >=3 Both odd Even/odd (2,3) >=3 Even/odd Both odd (5,6) ? Odd/even Even/odd (6,7) >=3 Both odd Even/odd (7,8) ? Even/odd Even/odd (10,11) ? Even/odd Even/odd (11,12) >=3 Even/odd Both odd (12,13) ? Odd/even Even/odd Table 3: Manhattan distance between consecutive colors for L(3,1,1) coloring.

(5,6), (7,8), (10,11) and (12,13) are the special cases we discussed in the previous text

(i.e. from L(2,1,1)). Each pair of colors has minimum distance at least 2. From the table 3

we can see that each pair of remaining colors has Manhattan Distance no less than 3.

According to table 1, their minimum distance should at least be 2. Therefore, (a) is

proved.

We see that for consecutive colors, their minimum distance is at least 2. It’s not difficult to prove minimum distance between each consecutive colors is exactly 2 by giving detailed examples, If the cellular grid is sufficiently large. In fact, by observing on from the figures 32, 31, it’s easy to see that each color is no more than 2 distance away from all other colors. Since we use the same tiling technique as L(2,1,1), the L(3,1,1)-

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coloring should have the same property as L(2,1,1)-coloring, that the minimum distance between each pair of consecutive colors is 2.

Now we check the color pairs (a,b), where |a-b|=2. We have the following table.

|a-b|=2 Manhattan Distance i Parity j Parity

(0,2) ? Even/odd Even/odd (1,3) ? Odd/even Even/odd (3,5) >=2 Both even Both odd (5,7) >=3 Odd/even Both odd (6,8) >=3 Even/odd Both even (8,10) >=2 Both even Both even (10,12) >=3 Both even Even/odd (11,13) >=3 Both odd Even/odd Table 4: Manhattan distance between colors (a,b), where |a-b|=2.

(0,2) (1,3) are the special cases we discussed in the previous text. Each pair of colors has minimum distance at least 2.

From the table we can see that (0,2), (1,3), (5,7), (6,8), (10,12) and (11,13) each

has Manhattan distance no less than 3. According to table 2, their minimum distance

should be at least two. For the remaining color pairs (3,5) and (8,10), each has Manhattan

distance no less than 2, but each has the same parity property for i or j co-ordinate of its

colors. As we have discussed in the previous text, their minimum distance is at least 2.

Therefore, (b) is proved.

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In addition, the Manhattan distance between two vertices of the same color is greater than or equal to 6. From f(u) we also know that all vertices of the same color have the same parity property for i or j coordinate. Then, it’s not difficult to see that the minimum distance between two vertices of the same color is greater than 3. So (c) is also proved.

Therefore the Function f (u) given by our algorithm for L(3,1,1) coloring is correct for coloring any size cellular graphs.

CHAPTER 4

Conclusion & Future Work

Here we discussed mainly the K chromatic coloring problem which considers only co-channel interference but not the adjacent channel interference. The coloring methods we discussed for the cellular graph networks are L(1), L(1,1), L(2,1), L(1,1,1) and L(2,1,1) from the literature. We gave the minimum number of colors required by the

L(3,1,1) coloring and also an algorithm for coloring a cellular graph network satisfying

L(3,1,1) condition. We showed that span (0, 13) is necessary and sufficient for L(3,1,1)- coloring of any cellular graph network. Here we get the minimized span but not the minimized order which is the lower boundary or the decision version of the problem. It would be nice to extend L(3,1,1)-coloring method to non static cellular graph networks, where not every cell has the same size. Even for static cellular graphs one can try to find the optimal coloring solution when adjacent channel interference is also considered.

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APPENDIX A

Frequency Assignment Tool

To help us in the frequency assignment for the L(3,1,1), we designed a software tool. The interface for the tool is given in the figure 34 below. In the interface, we have two input boxes to enter the number of rows and columns for the grid. After we have entered the number of rows and columns for the graph, we hit the graph generation button to generate the graph without any coloring (fig 35). Once we have the uncolored graph, we can select any coloring method we would like to apply to the graph, by pressing the appropriate button available. In fig 36 we color the graph in fig 35 by pressing the

L(3,1,1) coloring button. The entire color spectrum used for all the coloring methods is given in the interface. In order to start coloring a new graph, we have to enter the number of rows and columns and then hit the graph generation button and select the desired coloring method to color the graph.

Fig 34: Tool interface with number of rows and columns entered.

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Fig 35: Tool generating graph without coloring.

Fig 36: Tool coloring the graph with L(3,1,1) coloring method.

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Fig 37: Tool coloring a 25×30 grid using L(3,1,1) coloring method.

REFERENCES

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