9.8 Graph Coloring. Coloring Goal: Pick as few colors as possible so that two adjacent regions never have the same color. See handout.

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Presentation transcript:

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.