Graph Colouring
Assign colours so adjacent vertices differ.
Key idea
Assign colours so adjacent vertices differ.
In depth
A proper colouring assigns colours to vertices so that adjacent vertices differ. The chromatic number χ(G) is the fewest colours needed. Used in scheduling and map colouring.
Worked example
A triangle needs 3 colours; any bipartite graph needs 2.
Real-world uses
- Scheduling exams so students with shared subjects do not clash
- Assigning radio frequencies
Common mistake
Using too many colours without checking if fewer work.
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 Graph Colouring?
Assign colours so adjacent vertices differ. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Graph Colouring used in real life?
Real-world examples: Scheduling exams so students with shared subjects do not clash; Assigning radio frequencies.
What should I learn before Graph Colouring?
Make sure you are comfortable with Planar Graphs, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Graph Colouring?
Using too many colours without checking if fewer work.
How can I practise Graph Colouring?
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 Graph Colouring?
Move on to Four Colour Theorem 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.