Graph_Theory_Notes8.pdf

# Proof exercise plane triangulations maximal planar

• Notes
• 9

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Proof. Exercise! Plane triangulations (maximal planar graphs) Definition 5. A simple connected plane graph in which all faces have degree three is called a plane triangulation or triangulation . Lemma 1. A simple connected plane graph is a triangulation if and only if its dual is cubic. Every simple plane graph with order at least three is a spanning sub- graph of a triangulation. (Keep adding edges (but not vertices) while main- taining planarity until no further edge can be added.) On the other hand, no simple spanning supergraph of a triangulation can be planar. Thus triangulations are precisely maximal planar graphs .

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3 Euler’s formula Euler’s formula Denote by n ( G ) and m ( G ) the order and size of a graph G , respectively. If G is a plane graph, denote by f ( G ) its number of faces. Theorem 9 (Euler’s formula) . Let G be a connected plane graph with n vertices, m edges and f faces. Then n - m + f = 2 . Proof. We proceed by induction on f ( G ). If f ( G ) = 1, then every edge of G is a bridge, and so G is a tree as it is connected. Thus, m ( G ) = n ( G ) - 1, yielding n ( G ) - m ( G ) + f ( G ) = 2. Proof (continued). Suppose that the result is true for all connected plane graphs with less than f faces, where f 2, and let G be a connected plane graph with f faces. Choose an edge e of G that is not a bridge. Then G - e is a connected plane graph with n ( G - e ) = n ( G ) , m ( G - e ) = m ( G ) - 1, and f ( G - e ) = f ( G ) - 1 as the two faces of G separated by e are merged to form one face of G - e . By the induction hypothesis, n ( G - e ) - m ( G - e ) + f ( G - e ) = 2. Hence n ( G ) - m ( G ) + f ( G ) = 2. Corollary 1. All planar embeddings of a connected planar graph have the same number of faces. Proof. This follows immediately from Euler’s formula. Theorem 10. Let G be a simple planar graph with n 3 vertices and m edges. Then m 3 n - 6 . Moreover, m = 3 n - 6 if and only if every planar embedding of G is a triangulation. Proof. It suffices to prove the result for simple connected planar graphs. Let G be a simple connected planar graph with n 3 vertices and m edges. Let ˜ G be an planar embedding of G with f ( ˜ G ) faces. Since G is simple and connected with at least three vertices, deg( f ) 3 for all faces f of ˜ G . Therefore, 2 m = X f deg( f ) 3 f ( ˜ G ) = 3( m - n + 2) , or, equivalently, m 3 n - 6. Moreover, m = 3 n - 6 holds if and only if deg( f ) = 3 for all faces f of ˜ G , that is, if and only if ˜ G is a triangulation for every planar embedding ˜ G of G . Corollary 2. K 5 is nonplanar. This is because K 5 has n = 5 vertices and m = 10 > 3 n - 6 = 9 edges. Corollary 3. Every simple planar graph has a vertex of degree at most 5.
Note that 5 is best possible since there are 5-regular planar graphs (e.g. the icosahedron graph). Proof. Let G be a simple planar graph with n vertices and m edges. Then δ ( G ) average degree of G = v V ( G ) deg( v ) n = 2 m n 6 n - 12 n < 6 .

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