1. Graph Coloring
Lots of application – be it mapping routes, coloring graphs, building redundant systems, mapping genes, looking at traffic patterns (see the Philadelphia Broad Street example on page 371) …

2. Map Coloring What is the least number of colors needed to color a map?

3. Map Coloring Region  vertex Common border  edgeF I G H
Chromatic number χ(G)
What is the least number of colors needed to color the vertices of a graph so that no two adjacent vertices are assigned the same color?

4. Coloring the USA

5. Every planar graph is 4-colorable
Four color theorem
Every planar graph is 4-colorable
The proof of this theorem is one of the most famous and controversial proofs in mathematics, because it relies on a computer program. It was first presented in A more recent reformulation can be found in this article:
Formal Proof – The Four Color Theorem, Georges Gonthier, Notices of the American Mathematical Society, December 2008.

6. Do you always need four colors?
Four color theorem
Every planar graph is 4-colorable
Do you always need four colors?
000
001
010
100
110
011
111
101 K4

7. What about non-planar graphs?
Four color theorem
Every planar graph is 4-colorable
What about non-planar graphs?
K3,3 K5

8. Chromatic Numbers of Some Graphs
χ(G) = 1 iff ...
For Kn, the complete graph with n vertices, χ(Kn) = Corollary: If a graph has Kn as its subgraph, then χ(Kn)=
For Cn, the cycle with n vertices, χ(Cn) =
For any bipartite graph G χ(G) =
For any planar graph G, χ(G) ≤ 4 (Four Color Theorem)

9. Computing the Chromatic Number
There is no efficient algorithm for finding χ(G) for arbitrary graphs.
Most computer scientists believe that no such algorithm exists.
Greedy algorithm: sequential coloring:
Order the vertices in nonincreasing order of their degrees.
Scan the list to color each vertex in the first available color, i.e., the first color not used for coloring any vertex adjacent to it.
Questions:
Can you think of a graph for which ordering the vertices in nonincreasing order is significant?
Can you think of a graph for which this algorithm gives a less than optimal coloring?

10. Applications of Graph Coloring
map coloring
scheduling
eg: Final exam scheduling
Frequency assignments for radio stations
Index register assignments in compiler optimization

11. Example: Index Registers
source:
Another application: Let’s make our own exam schedule! Use this link to enter your information: