Unformatted text preview: ( iii ) Show that a connected simple planar graph has at least one vertex of degree at most 5. 4. Consider a connected simple planar graph with v ( ≥ 3) vertices, e edges and f regions. ( i ) Show that if e = 3 v6 then each region is a triangle. ( ii ) Deduce that a convex polyhedron with 12 vertices and 20 faces is composed entirely of triangles. 5. A graph is said to be polyhedral if it is simple, connected, and planar, and every vertex has degree at least 3. ( i ) Prove that a polyhedral graph cannot have exactly seven edges. ( ii ) Prove that no polyhedral graph has 30 edges and 11 regions. 6. Give an example of a connected planar graph in which e > 3 v6. 7. In this graph, ±nd a subgraph which is homeomorphic to K 3 , 3 . Is this graph planar or nonplanar?...
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 Spring '09
 Math, Graph Theory, Vertex, Planar graph, Glossary of graph theory, Hamiltonian path, Hamiltonian path problem

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