Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3. (i) Use Euler's formula to prove that G has a face bounded by at most four edges. (ii) Give an example to show that the result of part (i) is false if G has 12 faces.
Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3. (i) Use Euler's formula to prove that G has a face bounded by at most four edges. (ii) Give an example to show that the result of part (i) is false if G has 12 faces.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:Let G be a simple plane graph with fewer than 12 faces, in which each vertex has
degree at least 3.
(i) Use Euler's formula to prove that G has a face bounded by at most four edges.
(ii) Give an example to show that the result of part (i) is false if G has 12 faces.
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