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Graph Theory – Proofs that K5 and K3,3 are not planar www.waldomaths.com Copyright © R F Barrow 2009, all rights reserved The Proofs that K5 and K3,3 are not planar K5 Complete graph with 5 nodes K3,3 Complete bipartite graph with 3 + 3 nodes Proof that K5 is not planar Proof by contradiction Assume K5 is planar ---------------------- There’s a contradiction So K5 is not planar Proof that K5 is not planar Proof by contradiction Assume K5 is planar Then it obeys Euler’s Relationship R+N=A+2 N=5 A = 10 so 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 So K5 doesn’t obey Euler’s Relationship. So K5 is not planar QED Proof that K3,3 is not planar Proof by contradiction Assume K3,3 is planar Then it obeys Euler’s Relationship R+N=A+2 N=6 A=9 so 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 So K3,3 doesn’t obey Euler’s Relationship. So K3,3 is not planar QED A Waldomaths clip Copyright © R F Barrow 2009, all rights reserved this, along with other videos and more than 100 applets, all on mathematical themes, can be found on the free website www.waldomaths.com visit Waldomaths today