maths.free › Combinatorics & Graph Theory › Colouring and planar graphs

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.

ምሳሌ · 6 - 2

Arithmetic · Arithmetic

Evaluate 6 - 2

6 - 2

እርምጃ በደረጃ

  1. -2 + 6 = 4

    Add: -2 + 6 = 4.

መልሱን አሳይ
4

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: Colouring and planar graphs

  1. Add: -2 + 6 = 4.

ጥያቄዎች

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.

በ Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and Hamilton