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.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter1: Line And Angle Relationships
Section1.5: The Format Proof Of A Theorem
Problem 12E: Based upon the hypothesis of a theorem, do the drawings of different students have to be identical...
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