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Group no: 09
NAME :Tarafder,Md. Shakibuzzaman Id:13-23384-1
Feroz,Adnan Ahmed 13-22916-1
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.
Graph Coloring is an assignment of colors (or
any distinct marks) to the vertices of a graph.
Strictly speaking, a coloring is a proper coloring
if no two adjacent vertices have the same color.
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.
Vertex Coloring: It is a way of coloring the vertices of a
graph such that no two adjacent vertices share the
Edge Coloring: An edge coloring assigns a color to
each edge so that no two adjacent edges share the
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
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
λ = total no of colors
λ C= selecting I colors out of λ colors
i ΣCλ C= total number of ways a graph canbe properly
i colored using λ or lesser no. of colors
Pn(λ) of G = ΣCi
Let G be a simple graph, and let PG(k) be the number of ways
of coloring the verticles 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
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
If G is the complete graph Kn, then Pn(K) = K(K - 1)(K
- 2) . . . (K - n +1).
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.
Theorem - the vertices of every finite planar
graph can be coloured properly with five
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.
Mobile radio frequency assignment
Vertex coloring models to a number of scheduling problems .In the
cleanest form, a given set of jobs need to be assigned to time slots,
each job requires one such slot. Jobs can be scheduled in any order,
but pairs of jobs may be in conflict in the sense that they may not be
assigned to the same time slot, for example because they both rely on
a shared resource. The corresponding graph contains a vertex for every
job and an edge for every conflicting pair of jobs. The chromatic number
of the graph is exactly the minimum make span, the optimal time to
finish all jobs without conflicts.
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 Group Special 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
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
Solving Sudoku puzzles can be expressed as a graph coloring problem. Consider the
4 × 4 = 22 × 22 case. The aim of the puzzle in its standard form is to construct a proper
9-coloring of a particular graph, given a partial 4-coloring. The graph in question has
16 vertices, one vertex for each cell of the grid. The vertices can be labeled with the
ordered pairs (x, y), where x and y are integers between 1 and 4. In this case, two
distinct vertices labeled by (x, y) and (x′, y′) are joined by an edge if and only if:
x = x′ (same column) or,
y = y′ (same row) or,
⌈ x/2 ⌉ = ⌈ x′/2 ⌉ and ⌈ y/2 ⌉ = ⌈ y′/2 ⌉ (same 2 × 2 cell)
The puzzle is then completed by assigning an integer between 1 and 4 to each vertex,
in such a way that vertices that are joined by an edge do not have the same integer
assigned to them