1. Graph

2. Planar Graph
A graph is called planar if it can be drawn in the plane without any edges crossing
A graph may be planar even if it is usually drawn with crossings because it may be possible to draw it in a different way without crossings.

3. Is K4 planar?
K4 is planar because it can be drawn without crossings.
Is Q3 planar?
Q3 is planar, because it can be drawn without any edges crossing.

4. Graph Coloring
A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color.
A graph can be colored by assigning a different color to each of its vertices. What is the least number of colors necessary?
The chromatic number of a graph is the least number of colors needed for a coloring of this graph. denoted by X (G). (Here X is chi.)

5. What are the chromatic numbers of the graphs G and H
The chromatic number of G is at least three.
H has a chromatic number equal to 4.

6. What is the chromatic number of the complete bipartite graph Km,n , where m and n are positive integers?
Ans. X (Km ,n ) = 2
What is the chromatic no. of the graph Cn, where n >=3?
Ans. X (Cn ) = 2 if n is an even positive integer with n>= 4 and X (Cn ) = 3 if n is an odd positive integer with n>=3 .

7. Thank You