1. Introduction to Graph Theory
Lecture 12: Graph Coloring:
Vertex Coloring and Independent Sets

2. Introduction
Partition of the graph into disjoint sets such that items within a given set are mutually nonadjacent.
Applications including scheduling exams.
The vertices are courses
An edge represents sharing of students between two courses
These problems are related to graph coloring

3. The Chromatic Number
Vertex coloring of graph G is an assignment of clolors to the vertices so that adjacent vertices have distinct colors.
A graph that permits a k-coloring is called k-colorable.
The chromatic number of a graph G, , is the minimum number of colors needed for proper coloring.
What is the chromatic number for
A bipartite graph?
An odd cycle?

4. An Example
What is ? v8 v4 v6 v7 v2 v5 v1 v3

5. (con’t) If G is k-colorable means that
To prove that , we must show that
and
it is possible to k-color G.
Show that

6. K-critical G graph is k-critical if
, and
for every vertex
If then G must contain a k-critical subgraph
Why?
Theorem 6.1: If G is k-critical, then

7. Proof for Theorem 6.1 Done by contradiction Let G be k-critical, where
Let v be a vertex of G satisfying
We know that removing v makes
Since , there is one color, say r, among the k-1 colors used to color G-v not used by any neighbors of v. So color v with r.
This implies G is (k-1)-colorable => contradiction!

8. A related theorem
Theorem 6.2: If , then G must have at least k vertices of degree at least k-1.
Proof:
We know that G has a k-critical subgraph H.
It implies that H has at least k vertices (every vertex has at least one vertex).
Since , H has at least k vertices of degree at least k-1.

9. Independence
A subset S of vertices are independent if no pair of vertices of S are adjacent.
So the vertices of the same color form an independent set
Can you find an independence set in the graph b f g a h d c j e i

10. Independence Number : the max size of an independent set
What is for our previous graph?
How about and ?
We have a very nice lower bound for
The upper bound is

11. The Lower Bound for Since , V(G) can be partitioned into k color sets
Since all vertices of cannot be adjacent with one another, therefore each is independent, and we have
So,

12. The Upper Bound for Let S be the largest indep. Set. So
Now let H be the graph with vertices V(G)-S.
It follows that
Since , we have
Combining the two inequalities we get

13. Uniquely k-Colorable Graph
A partition of a set X is a collection of k disjoint subsets.
In coloring, V(G) can be partitioned into k subsets, and no pair of adjacent vertices of G are in the same subset.
How many ways can we partition ?
The existence of more than one partition is not always possible. For example

14. Uniquely k-Colorable Graph
The existence of more than one partition is not always possible. For example a e b c d

15. Uniquely k-Colorable Graph
A graph G with is uniquely k-colorable if there is only one partition of V(G) into k subset.
Theorem 6.3: If G is uniquely k-colorable, then
Proof: (done w/o using the fact that G is k-critical)
By contradiction again.

16. (cont) Let the vertices of G be colored using
If a vertex v is assigned , it must be adjacent to every other color
If this is not true, then we can change the color of v to the color not used by its neighbor, thus changed the partition
This contradicts the fact that G is uniquely k-colorable.