Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Graph Theory – Proofs that K5 and K3,3 are not planar Copyright © R F Barrow 2009, all rights reserved www.waldomaths.com."— Presentation transcript:

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

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 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 Copyright © R F Barrow 2009, all rights reserved visit Waldomaths today


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

Similar presentations


Ads by Google