Discrete mathematics and graph theory · Maths EE idea · Ambitious

Why five colours are enough

A research question to start from

How does the five-colour theorem for planar maps follow from Euler's formula, and why is the four-colour case so much harder?

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Why it works as a maths EE

A complete proof you can own, set against the famous computer-assisted proof.

Mathematics you would need

Planar graphs
Euler's formula and degree bounds
Kempe chains
Proof by induction

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Prove every planar graph has a vertex of degree at most 5.
Prove the six-colour then five-colour theorems.
Explain where Kempe's four-colour argument fails.

Scope and difficulty

Ambitious. Ambitious but bounded.

Pitfalls

History of the four-colour proof.
Gaps in the Kempe-chain step.

Where to start reading

Search a library catalogue or a university's open lecture notes for: five colour theorem proof Kempe chains; Kempe's error. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

Plan this EEOpens the free EE planner with this question as a starting point. Sharpen the questionSix quick checks for a focused research question. Self-check a draftCheck a section against each criterion, free.