GRAPH COLORING

Submitted By:

Rashika Ahuja:110101203

Sachin Yadav:110101210

Shadab Masoodi:110101224

INTRODUCTION In graph theory, graph coloring is a special case of graph labeling.

It is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints.

ORIGIN Coloring theory started with the problem of coloring the countries of a map in such a

way that no two countries that have a common border receive the same color.

If we denote the countries by points in the plane and connect each pair of points that correspond to countries with a common border by a curve, we obtain a planar graph.

Graphs are used to depict ”what is in conflict with what”, and colors are used to denote the state of a vertex.

So, more precisely, coloring theory is the theory of ”partitioning the sets having Internal unreconcilable conflicts.

TERMINOLOGY Vertex Coloring: It is a way of coloring the vertices of a graph such that no two adjacent

vertices share the same color.

Edge Coloring: An edge coloring assigns a color to each edge so that no two adjacent edges share the same color.

Face Coloring : A face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Chromatic Number: The chromatic number of a graph is the minimum number of colors in a proper coloring of that graph. If chromatic number is r then the graph is r-chromatic.

Chromatic number: 4

Chromatic polynomial

Polynomial which gives the number of ways of proper coloring a graph using a given number of colors

Ci = no. of ways to properly color a graph using exactly i colorsλ = total no of colors λ Ci = selecting I colors out of λ colors ΣCi

λ Ci = total number of ways a graph canbe properly colored using λ or lesser no. of colors

Pn(λ) of G = ΣCi λ Ci

C1 =0 C2=0C3 =C4=

P4 (λ) of G = C1(λ) + C2(λ) (λ-1)/2! + C3(λ) (λ-1) (λ-2) /3! + C4(λ) (λ-1) (λ-2) (λ- 3)/4!

K-chromatic Graph

Let G be a simple graph, and let PG(k) be the number of ways of coloring the vertices of G with k

colors in such a way that no two adjacent vertices are assigned the same color. The function PG(k) is

called the chromatic polynomial of G.As an example, consider complete graph K3 as shown in the following figure.

 

  

Then the top vertex can be assigned any of the k colors, the left vertex can be assigned any k-1 colors, and right vertex can be assigned any of the k-2 colors.

The chromatic polynomial of K3 is therefore K(K -1)(K -2). The extension of this immediately gives us the following result.

If G is the complete graph Kn, then Pn(K) = K(K - 1)(K - 2) . . . (K - n +1).

OBSERVATIONS Every non-trivial graph is atleast 2-chromatic.

If a graph has a triangle in it , then it is atleast 3-chromatic.

Chromatic Polynomial for a tree :

Pn(λ) of Tn = (λ) (λ-1)n-1 (tree is 2-chromatic)

This can be proved by Mathematical Induction.

Tree is 2-chromatic.

The five colour problem

Theorem - the vertices of every finite planar graph can be coloured properly with five colours.

Proof-the proof is based on induction on vertices of a planar graph, since the vertices of all planar graph G with 1,2,3,4,5 can be properly coloured by 5 or less colours.

Let us assume that every planar graph with n-1 vertices is properly colourable with 5 colours or fewer. So we have to show that there is no graph of n-vertices which require more than 5-colours for proper colouring.

FIVE COLOR THEOREM

APPlCATIONS Sudoku

Scheduling

Mobile radio frequency assignment

Pattern matching

Register Allocation

Mobile Radio Frequency Assignment  When frequencies are assigned to towers, frequencies assigned to all towers at the

same location must be different. How to assign frequencies with this constraint? What is the minimum number of frequencies needed? This problem is also an instance of graph coloring problem where every tower represents a vertex and an edge between two towers represents that they are in range of each other.

 GSM (Global System for Mobile Communications, originally Groupe Spécial Mobile), was created in 1982 to provide a standard for a mobile telephone system.. Today, GSM is the most popular standard for mobile phones in the world, used by over 2 billion people across more than 212 countries.

GSM is a cellular network with its entire geographical range divided into hexagonal cells.

Each cell has a communication tower which connects with mobile phones within the cell.

All mobile phones connect to the GSM network by searching for cells in the immediate vicinity.

GSM networks operate in only four different frequency ranges. The reason why only four different frequencies suffice is clear: the map of the cellular regions can be properly colored by using only four different colors! So, the vertex coloring algorithm may be used for assigning at most four different frequencies for any GSM mobile phone network