Kempe's universality theorem

Imagine you have lots of different shapes, like circles, squares, and triangles, and you want to color them all in a special way so that no two shapes that touch each other have the same color. This is called a "coloring problem."

Kempe's Universality Theorem is a really cool idea that helps us solve this coloring problem for certain shapes called graphs. A graph is a collection of points, called "vertices," that are connected by lines or curves, called "edges."

Here's how Kempe's Universality Theorem works: If you have a graph that can't be colored the way we want, we can actually transform it into a different graph that can be colored the way we want! This transformation is called a "Kempe chain."

A Kempe chain is like a path through the graph that helps us switch the colors of some of the vertices in a certain way. By using Kempe chains, we can change the original coloring problem into a new one that's easier to solve. And if we keep doing this over and over again, we eventually end up with a graph that we can color perfectly.

So basically, Kempe's Universality Theorem helps us solve a tricky coloring problem by finding a way to transform it into an easier coloring problem using Kempe chains.