Upcoming SlideShare
×

# Graph T

• 250 views

• Comment goes here.
Are you sure you want to
Be the first to comment
Be the first to like this

Total Views
250
On Slideshare
0
From Embeds
0
Number of Embeds
0

Shares
2
0
Likes
0

No embeds

### Report content

No notes for slide

### Transcript

• 1. Graph Theory through an example of Map Coloring
• 2. Look at the map of an Island with 6 regions
A, B, C, D, E, F
Figure 1
• 3. We want to give
a unique color to each region.
This can be put in mathematical terms as follows:
V = {A, B, C, D, E, F},
C – a set of colors, f: V C
We callfa coloring of the map.
For obvious reasons, we decide to give distinct colors to two neighboring regions.
Iff satisfies this condition, we call it aproper coloring.
• 4. Now, two questions arise:
What is the minimum number of colors required for a proper coloring?
With a certain number of colors we have, in how many ways can a proper coloring be done?
For the given map with only 6 regions it may be easy to answer these questions.
We can make it easy for any map as follows.
• 5. For the given map,
represent each region by a point.
If two regions are neighboring,
draw a line segment joining them.
In the resulting figure,
we refer to the points asverticesand
the line segments asedges.
.
.
.
.
F
D
B
A
.
.
C
E
Figure 2
• 6. Figure 2 represents what we call agraph in mathematics.
Let us have its formal definition:
V – a finite set, E – a set of pairs of elements in V
G = (V, E) is called a graph.
Elements of V are called vertices and
those of E are called edges.
If {a, b} ε E, we say the vertices a and b are adjacent.
Now, the question of coloring
the regions of the map
becomes that of allotting a color to
each vertex of the graph G.
Map coloring becomes graph coloring.
• 7. In Figure 2
the graph is G = (V, E), where
V = {A, B, C, D, E, F}
E = {AB, AC, BC, BD, BE, CE, DE, DF, EF},
Where AB stands for the edge {A, B}.
• 8. Graph Coloring
If G = (V, E) is a graph and C is a set of colors,
then a function f: V  C is called
a coloring of the graph.
It is a proper coloring if
The least number of colors
required for a proper coloring is called
the chromatic number of the graph,
denoted by (G).
If  is the number of colors available,
the number of possible proper colorings will be a polynomial, denoted by p(G, ) – the chromatic polynomial.
• 9. Two Special Cases:
G = (V, E), E =  - null graph
G = (V, E), E is the set of all pairs of elements in V – complete graph.
If |V| = n, the complete graph is denoted by Kn
• 10. Examples
Null Graph: G = (V, E), |V| = n, E = p(G, ) = n(G) = 1
Complete Graph: G = Kn p(G, ) = (n) = ( - 1)( - 2)…( - n + 1)(G) = n
Linear Graph: G = (V, E), V = {v1, v2,…,vn}, E = {{vi, vi+1}, i = 1, 2, …, n-1}p(G, ) = ( -1)n-1(G) = 2
• 11. Union and Intersection of graphs
• 12. Example 1
Consider the graph G = (V, E),
V = {A, B, C, D}, E = { AB, AC, AD, BC, CD}
G1 = (V1, E1), V1 = {A, B, C}, E1 = {AB, AC, BC}
G2 = (V2, E2), V2= {A, D, C}, E2 = {AD, AC, DC}
G is the union of G1 and G2
Their intersection is H = ({A, C}, {AC})
.
.
D
C
.
.
A
B
Figure 4
• 13. Theorem
If G = (V, E) is a graph and G1, G2 are its subgraphs such that G = G1G2, G1G2 = Kn, then
• 14. Example 2
In Example 1
G1 = K3, G2 = K3
G1G2 = K2
Therefore,
Here p(G, 1) = p(G, 2) = 0. What does this mean?
And p(G, 3) = 6
So, the chromatic number (G) = 3
There are 6 proper colorings possible with 3 colors.
• 15. Example 3
Now, let us take up the initial problem of map coloring.
In the corresponding graph,
let V1 = {A, B, C, E},
E1 = {AB, AC, BC, BE, CE}
V2 = {B,D,E,F},
E2 = {BD, BE, DE, DF, EF}
G1 = (V1, E1), G2 = (V2, E2)
G = G1G2, G1G2 = K2
• 16. By Example 2,
p(G1, ) = p(G2, ) = (-1)(-2)2
Also, p(K2, ) = (-1)
Thus, (G) = 3, and there are 6 proper colorings possible with 3 colors.
.
.
.
.
F
D
B
A
.
.
C
E
• 17. A Proper Coloring of the Map