Four-Color Theorem

Definition

The Four-Color Theorem states that any map drawn on a plane can be colored using no more than four colors, such that no two adjacent regions share the same color. This theorem is a fundamental result in graph theory and topology.

5 Must Know Facts For Your Next Test

1. The theorem was first conjectured in 1852 by Francis Guthrie.
2. It was proven using computer assistance by Kenneth Appel and Wolfgang Haken in 1976.
3. The proof involves checking a large number of configurations to ensure they adhere to the four-color rule.
4. This theorem applies not only to geographic maps but to any planar graph representation.
5. It has significant implications in network design, scheduling problems, and other areas requiring graphical optimization.

Review Questions

• What does the Four-Color Theorem state about coloring maps?
• Who were the mathematicians responsible for proving the Four-Color Theorem?
• In what year was the Four-Color Theorem proven with computer assistance?

Related terms

Planar Graph: A graph that can be embedded in the plane so that no edges intersect except at their endpoints.

Graph Coloring: An assignment of labels (colors) to each vertex of a graph such that no two adjacent vertices share the same color.

Chromatic Number: The smallest number of colors needed to color a graph according to its color constraints.