Ringel

Ringel may refer to several different things, depending on the context:

• Gerhard Ringel (1919-2008): A German mathematician known for his work in graph theory, particularly his contributions to the Heawood map-coloring problem and the Ringel-Youngs theorem, which determines the genus of complete graphs. He also worked on problems in combinatorial enumeration.

• Ringel-Youngs Theorem: A fundamental result in topological graph theory. It provides a formula for calculating the genus of a complete graph Kn. The theorem states that the genus γ(Kn) required to embed Kn without crossings is given by γ(Kn) = ⌈(n − 3)(n − 4)/12⌉, where ⌈x⌉ denotes the ceiling function (the smallest integer greater than or equal to x). This theorem provides a complete solution to the Map Color Theorem, originally conjectured by Heawood.

• Ringel Array: A specific type of Latin square used in designing statistical experiments, particularly in situations requiring near balance.

• Ringel's Map Problem: This is another name for the Heawood map-coloring problem. The problem asks for the minimum number of colors needed to color any map on a given surface such that no two adjacent regions share the same color. Ringel made significant contributions to its solution.

Therefore, when encountering the word "Ringel", it is essential to consider the surrounding context to determine the specific meaning being referred to, most commonly in mathematical or graph-theoretical discussions.