h6a - MATH 4171 Graph Theory Homework Set VI -Solutions...

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MATH 4171 Graph Theory SPRING 2006 Homework Set VI -Solutions 1. Let G = ( V, E ) be a plane graph of minimum degree at least three. Prove that G has a region of size at most Fve. Proof. Let d 1 , d 2 , ..., d | V | be the degrees of G . Then 2 | E | = d 1 + d 2 + ... + d | V | 3 | V | , so | V | ≤ 2 | E | / 3. Let s 1 , s 2 , ..., s r be the sizes of regions of G . Then 2 | E | = s 1 + s 2 + ... + s r . If all regions have size at least six, then 2 | E | ≥ 6 r , so r ≤ | E | / 3. However, we deduce from Euler formula that 2 = | V | - | E | + r 2 3 | E | - | E | + 1 3 | E | = 0, a contradiction, which proves that s i 5 for at least one i . 2. Decide if the two given graphs are planar. You need to justify your conclusions. Solution. The Frst graph is planar. A redrawing of this graph is shown on the right. The second graph is non-planar, because deleting the two center vertices results in a subdivision of K 3 , 3 . x
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This note was uploaded on 02/17/2009 for the course MATH 4171 taught by Professor Lax,r during the Spring '08 term at LSU.

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