Math 2.0

Combinatorics Theorem

Edge bound for planar graphs

Builds on Euler's formula for planar graphs

Statement

Let GG be a finite simple connected planar graph with v≥3v \ge 3 vertices and ee edges. Then e≤3v−6e \le 3v - 6.

Remarks

Proof idea: each face is bounded by at least 33 edges, and each edge touches at most 22 faces. Thus 3f≤2e3f \le 2e, and Euler's formula gives the bound. If GG also has no cycle of length 33, then e≤2v−4e \le 2v - 4. A consequence: each finite simple planar graph has a vertex of degree at most 55.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Graph, definition, no evidence levelGraphdefinitionPlanar graph, definition, no evidence levelPlanar graphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionConnected graph, definition, no evidence levelConnected graphdefinitionEuler's formula for planar graphs, statement, no evidence levelEuler's formula for p…statementEdge bound for planar graphs, statement, no evidence levelEdge bound for planar…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Planar graph definition
  3. Walk definition
  4. Reachability definition
  5. Connected graph definition
  6. Euler's formula for planar graphs statement
  7. Edge bound for planar graphs statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Planar_graph, Euler's formula
About this page
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Kindstatement, theorem
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UsesEuler's formula for planar graphs

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