This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.

1. Submitted By:
Rashika Ahuja:110101203
Sachin Yadav:110101210
Shadab Masoodi:110101224

2.  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.

3.  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.

6.  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.

7.  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.

8.  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.

9.  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 rchromatic.
Chromatic number: 4

10.  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
colors
 λ = 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

11.  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!

12. 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.

13.  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).

14.  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.

15. 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.

17.  Sudoku
 Scheduling
 Mobile radio frequency assignment
 Pattern matching
 Register Allocation

18.  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.

19.  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.

21.  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