Download presentation

Presentation is loading. Please wait.

Published byLillie Wisbey Modified over 2 years ago

1
Graph Theory – Proofs that K5 and K3,3 are not planar Copyright © R F Barrow 2009, all rights reserved www.waldomaths.com

2
K5K5 K 3,3 The Proofs that K 5 and K 3,3 are not planar Complete graph with 5 nodes Complete bipartite graph with 3 + 3 nodes

3
Proof that K 5 is not planar Proof by contradiction Assume K 5 is planar So K 5 is not planar ---------------------- There’s a contradiction

4
Proof that K 5 is not planar Proof by contradiction Assume K 5 is planar Then it obeys Euler’s Relationship So K 5 doesn’t obey Euler’s Relationship. So K 5 is not planar R + N = A + 2 N = 5A = 10so R = 7 Each region is bounded by at least three arcs So there are at least 3R region boundaries So there are at least 3R ÷ 2 arcs, since each arc is 2 region boundaries 3 × 7 ÷ 2 = 10.5, so at least 11 arcs But there are 10 arcs There’s a contradiction QED

5
Proof that K 3,3 is not planar Proof by contradiction Assume K 3,3 is planar Then it obeys Euler’s Relationship So K 3,3 doesn’t obey Euler’s Relationship. So K 3,3 is not planar R + N = A + 2 N = 6A = 9so R = 5 Each region is bounded by at least four arcs So there are at least 4R region boundaries So there are at least 4R ÷ 2 or 2R arcs, since each arc is 2 region boundaries 2 × 5 = 10, so at least 10 arcs But there are 9 arcs There’s a contradiction QED

6
A Waldomaths clip this, along with other videos and more than 100 applets, all on mathematical themes, can be found on the free website www.waldomaths.com Copyright © R F Barrow 2009, all rights reserved visit Waldomaths today

Similar presentations

OK

Upper and Lower Bounds 2 - Calculations www.waldomaths.com Copyright © Waldomaths.com 2010, all rights reserved.

Upper and Lower Bounds 2 - Calculations www.waldomaths.com Copyright © Waldomaths.com 2010, all rights reserved.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on leverages statistics Ppt on uti mutual funds Writing process elementary ppt on cells Ppt on forest and wildlife heritage of india Ppt on power system transient stability Ppt on ministry of corporate affairs csr Ppt on vitamin d Ppt on company act 1956 provisions Ppt on area of parallelogram Ppt on sources of energy for class 8th math