1) Graph coloring is assigning colors to graph vertices such that no adjacent vertices have the same color. The minimum colors needed is the chromatic number. 2) Common graph types and their chromatic numbers are: cycle graphs have 2 or 3 colors, planar graphs have at most 4 colors, complete graphs need as many colors as vertices. 3) Cut vertices and edges, when removed, disconnect a graph into multiple components. Edge and vertex connectivity refer to the minimum numbers of edges or vertices that must be removed to disconnect a graph.

1. Connectivity and colored
graph
By students:
Hujeen yousif
Huveen husain
Zozan manaf
Supervisor: Dr. Didar A. Ali

2.  Graph Coloring
Graph Coloring is a process of assigning colors to the vertices
of a graph
such that no two adjacent vertices of it are assigned the same
color.
 Graph Coloring is also called as Vertex Coloring.
 It ensures that there exists no edge in the graph whose end vertices are colored
with the same color.
 Such a graph is called as a Properly colored graph.

3. Graph Coloring Example:-
The following graph is an example of a properly colored graph-
In this graph,
 No two adjacent vertices are colored with the same color.
 Therefore, it is a properly colored graph.
1
2
3
4
5

4.  Chromatic Number:-
Chromatic Number is the minimum number of colors required to properly color any
graph.
OR
Chromatic Number is the minimum number of colors required to color any graph
such that no two adjacent vertices of it are assigned the same color.

5. Chromatic Number Example:-
Consider the following graph-
In this graph,
 No two adjacent vertices are colored with the same color.
 Minimum number of colors required to properly color the vertices = 3.
 Therefore, Chromatic number of this graph = 3.
 We can not properly color this graph with less than 3 colors.
1
2
4 3
5

6.  Chromatic Number Of Graphs:-
Chromatic Number of some common types of graphs are as
follows-
1.Cycle Graph-
 a simple graph of ‘n’ vertices (n>=3) and ‘n’ edges forming
a cycle of length ‘n’ is called as a cycle graph.
 in a cycle graph, all the vertices are of degree 2some
 Chromatic Number
 If number of vertices in cycle graph is even, then its
chromatic number = 2.
 If number of vertices in cycle graph is odd, then its
chromatic number = 3.

7. Examples:-
.
1
2
3
1
2
3

8. 2. Planar Graphs:-
A Planar Graph is a graph that can be drawn in a plane such
that none of its edges cross each other.
Chromatic Number
Chromatic Number of any Planar Graph
= Less than or equal to 4

9. Examples:-
1
2
3
2 1
4
3

10. 3. Complete Graphs:-
 complete graph is a graph in which every two distinct vertices are joined by
exactly one edge.
 In a complete graph, each vertex is connected with every other vertex.
 So to properly it, as many different colors are needed as there are number of
vertices in the given graph.
Chromatic Number
Chromatic Number of any Complete Graph
=Number of vertices in that Complete Graph

11. Examples:-

12. 4. Bipartite Graph:-

13. 5. Trees:-
 A Tree is a special type of connected graph in which there are no circuits.
 Every tree is a bipartite graph.
 So, chromatic number of a tree with any number of vertices = 2.
Chromatic Number
Chromatic Number of any tree
=
2

14. Examples:-

15. Example:-
Find chromatic number of the following graph-
solution:
 From here,
 Minimum number of colors used to color the given graph are 2.
 Therefore, Chromatic Number of the given graph = 2.


16.  The given graph may be properly colored using 2
colors as shown below -

17. • Cut Vertex
Let ‘G’ be a connected graph. A vertex V ∈ G is called a cut vertex of ‘G’, if ‘G-V’
(Delete ‘V’ from ‘G’) results in a disconnected graph. Removing a cut vertex from a
graph breaks it in to two or more graphs.
 Note − Removing a cut vertex may render a graph disconnected.
 A connected graph ‘G’ may have at most (n–2) cut vertices.

18. • Example
In the following graph, vertices ‘e’ and ‘c’ are the cut vertices.
By removing ‘e’ or ‘c’, the graph will become a
disconnected graph.

19. there is no path between vertex ‘c’ and vertex ‘h’ and
many other. Hence it is a disconnected graph with cut
vertex as ‘e’. Similarly, ‘c’ is also a cut vertex for the
above graph.

20. • Cut Edge (Bridge)
Let ‘G’ be a connected graph. An edge ‘e’ ∈ G is called a cut edge if ‘G-e’ results in a
disconnected graph.
If removing an edge in a graph results in to two or more graphs, then that edge is
called a Cut Edge.
 Example
In the following graph, the cut edge is [(c, e)].

21. By removing the edge (c, e) from the graph, it becomes a disconnected graph.
In the above graph, removing the edge (c, e) breaks the graph into two which is
nothing but a disconnected graph. Hence, the edge (c, e) is a cut edge of the graph

22. Note − Let ‘G’ be a connected graph with ‘n’ vertices, then
a cut edge e ∈ G if and only if the edge ‘e’ is not a part of any cycle in G.
the maximum number of cut edges possible is ‘n-1’.
whenever cut edges exist, cut vertices also exist because at least one vertex of a cut
edge is a cut vertex.
if a cut vertex exists, then a cut edge may or may not exist.

23. • Edge Connectivity
Let ‘G’ be a connected graph. The minimum number of edges whose removal
makes ‘G’ disconnected is called edge connectivity of G.
 Notation − λ(G)
In other words, the number of edges in a smallest cut set of G is called the edge
connectivity of G.
If ‘G’ has a cut edge, then λ(G) is 1. (edge connectivity of G.)

24. • Example
Take a look at the following graph. By removing two minimum edges, the connected
graph becomes disconnected. Hence, its edge connectivity (λ(G)) is 2
Here are the four ways to disconnect the graph by removing two edges −

25. Vertex Connectivity
Let ‘G’ be a connected graph. The minimum number of vertices whose removal
makes ‘G’ either disconnected or reduces ‘G’ in to a trivial graph is called its
vertex connectivity.
Notation − For any connected graph G,
K(G) ≤ λ(G) ≤ δ(G)
Vertex connectivity (K(G)), edge connectivity (λ(G)), minimum number of degrees
of G(δ(G)).

26. Reference
 https://www.tutorialspoint.com/graph_theory/graph_theory_connectivity.htm
 https://www.gatevidyalay.com/graph-coloring-chromatic-number/