Math 2.0

Combinatorics Planar graphs, embeddings, minors Theorem

Euler's formula for planar graphs

Builds on Connected graphPlanar graph

Statement

Let GG be a finite connected planar graph, drawn in the plane with no crossing edges. Let vv, ee and ff be the numbers of vertices, edges and faces of the drawing, where the unbounded face counts. Then v−e+f=2v - e + f = 2.

Remarks

Proof idea: if the graph has a cycle, delete an edge of the cycle. This removes one edge and one face. At the end you have a spanning tree, with e=v−1e = v - 1 and f=1f = 1. The same formula holds for the vertices, edges and faces of a convex polyhedron.

Reading path

To understand this page from the ground up, read these 6 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…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.
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  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

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_characteristic, Plane graphs
About this page
Tag0W1C598HMS
URIm2://0W1C598HMS
Kindstatement, theorem
WritingW0 stub
MSC05C10
StatusThis copy of the site has no build data. Run m2 status 0W1C598HMS.
UsesConnected graph, Planar graph

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