Four Colour Theorem
Every planar map is 4-colourable.
Key idea
Every planar map is 4-colourable.
In depth
Every planar graph can be coloured with at most four colours. It was first proved in 1976 by Appel and Haken with computer assistance, the first major theorem proved that way.
Worked example
Any map of countries on a plane needs no more than four colours.
Real-world uses
- Colouring political maps with only four colours so neighbouring regions differ
Common mistake
Believing the theorem applies to maps with disconnected regions or on a torus.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Four Colour Theorem?
Every planar map is 4-colourable. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Four Colour Theorem used in real life?
Real-world examples: Colouring political maps with only four colours so neighbouring regions differ.
What should I learn before Four Colour Theorem?
Make sure you are comfortable with Graph Colouring, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Four Colour Theorem?
Believing the theorem applies to maps with disconnected regions or on a torus.
How can I practise Four Colour Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Four Colour Theorem?
Move on to Bipartite Graphs and Matching next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.