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t04_y2008 - iii Show that a connected simple planar graph...

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The University of Sydney MATH2969/2069 Graph Theory Tutorial 4 (Week 11) 2008 1. ( i ) How many Hamiltonian cycles are there in this graph? X 7 6 7 9 9 5 6 9 7 8 ( ii ) Delete the vertex labelled X (and its incident edges). How many spanning trees are there on the remaining subgraph? How many of these spanning trees are part of a Hamiltonian cycle? ( iii ) Find a minimum weight spanning tree on the remaining subgraph. Hence show that the weight (or length) of a solution to the travelling salesman problem is at least 32. 2. Show that this graph is planar by drawing it in the plane without any edges crossing. Verify Euler’s formula for this graph. 3. ( i ) Show that a connected simple planar graph all of whose vertices have degree at least 5 must have at least 12 vertices. ( ii ) Show that a connected simple planar graph with fewer than 30 edges has at least one vertex of degree at most 4.
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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 v-6 then each region is a triangle. ( ii ) Deduce that a convex polyhedron with 12 vertices and 20 faces is com-posed 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 v-6. 7. In this graph, ±nd a subgraph which is homeomorphic to K 3 , 3 . Is this graph planar or nonplanar?...
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