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Colouring and planar graphs

Chromatic number, Euler's formula V − E + F = 2, and the four-colour theorem.

Colour vertices so neighbours differ; the fewest colours needed is the chromatic number. For a planar graph V − E + F = 2, which bounds edges by 3V − 6 and shows K₅ is not planar. Picture it: any map needs at most four colours, proven with a computer in 1976. Think it: Euler's formula is the Euler characteristic, the same invariant that distinguishes a sphere from a torus.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 6 - 2

Arithmetic · Arithmetic

Evaluate 6 - 2

6 - 2

Àwọn ìṣàmúlò-ètò

-2 + 6 = 4
Add: -2 + 6 = 4.

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Bii o ṣe le: Colouring and planar graphs

Add: -2 + 6 = 4.

Àwọn Àtòjọ-ẹ̀yàn

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Diẹ̀ nínú Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Recurrences and generating functions Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton