Maximum of Seven Colors Needed for Proper Vertex Coloring on Torus

Theorem

Let $G$ be a graph embedded on the surface of a torus.

$G$ can be assigned a proper vertex $k$-coloring such that $k \le 7$.

Proof

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Historical Note

This result was known before the Four Color Theorem, its counterpart for planar graphs.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): four-colour problem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): four-colour problem