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Euler’s theorem and applications Martin B ODIN martin.bodin@ens-lyon.org Euler’s theorem and applications – p. 1
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The theorem Euler’s theorem and applications – p. 2
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The theoremTheorem. Given a plane graph, if v is the number of vertex,e, the number of edges, and f the number of faces, v−e+f =2 Euler’s theorem and applications – p. 2
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The TheoremProof. Consider the plane graph G. Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.We consider T , a minimal graph from G, connex. Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.We consider T , a minimal graph from G, connex.T is a tree.Thus eT = v − 1, where eT is the number of T ’s edge. Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.Then we consider the dual graph. Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.Then we consider the dual graph.And the dual D of T . Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.Then we consider the dual graph.And the dual D of T .D in a also a tree.Thus eD = f − 1. Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.Now, we have eT + eD = e. e = (v − 1) + (f − 1) Euler’s theorem and applications – p. 3
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The TheoremProof. Consider the plane graph G.Now, we have eT + eD = e. v−e+f =2 Euler’s theorem and applications – p. 3
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Applications Euler’s theorem and applications – p. 4
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ApplicationsGiven a plane graph, there exists an edge ofdegree at more 5. Euler’s theorem and applications – p. 4
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ApplicationsGiven a plane graph, there exists an edge ofdegree at more 5.Given a ﬁnite set of points non all in the sameline, there exists a line that contains only two ofthem. Euler’s theorem and applications – p. 4
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Thanks For Your Listenning !Any questions ? Euler’s theorem and applications – p. 5
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