9.8 Graph Coloring
Coloring
Goal: Pick as few colors as possible so that two
adjacent regions never have the same color.
See handout
Dual graph
Vertex- used for a region
Edge- used if the regions represented by the two
vertices have a common border
Note: Any map in the plane has a planar dual
graph.
Q: how many colors are necessary?
Coloring definitions
Def: A coloring of 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.
The chromatic number of a graph is the least
number of colors needed.
Handout- coloring activity
3, 4,… colors
Q: Construct a graph that needs 3 colors, 4
colors, …
Examples…
Four Color Theorem
Four Color Theorem: The chromatic number of a planar
graph is no larger than 4.
Note: This was an open problem for 120 years.
Q: Is it possible to find a graph with a chromatic number
greater than 4?
Chromatic number of Km,n
Ex: Chromatic number of Km,n?
chromatic number of 2
Claim: A simple graph with a chromatic number
of 2 is bipartite.
Proof:
Chromatic number of Cn
• Ex: Chromatic number of Cn when n is odd.
• Ex: Chromatic number of Cn when n is even.
4 color theorem- US map
Applications
Ex. 1: Final Exams scheduling
Goal: no student has 2 final at same time
Use an edge if …
Ex. 2: class scheduling
Use an edge if two classes may have students in common.
…
Ex. 3: no faculty member has two committee
meetings at the same time
…
• Ex 4: channel assignments, where no two
stations within 150 miles of each other can
have the same channel.
• Edges connect vertices if the sites are located
within 150 miles.