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Unformatted text preview: The University of Sydney MATH2969/2069 Graph Theory Tutorial 5 (Week 12) 2008 1. ( i ) Let G be the disconnected planar graph shown. Draw its dual G * , and the dual of the dual ( G * ) * . ( ii ) Show that if G is a disconnected planar graph, then G * is connected. Deduce that ( G * ) * is not isomorphic to G . G 2. A certain polyhedron has faces which are triangles and pentagons, with each triangle surrounded by pentagons and each pentagon surrounded by triangles. If every vertex has the same degree, p say, show that 1 e = 1 p- 7 30 . Deduce that p = 4, and that there are 20 triangles and 12 pentagons. Can you construct such a polyhedron? State the dual result. 3. Determine the chromatic number of each of the following graphs: a b c d e f g ( i ) a b c d e f g h i j k ( ii ) a b c d e f g h i j k l ( iii ) a b c d e f g ( iv ) 4. For each of the following graphs, what does Brooks Theorem tell you about the chromatic number of the graph? Find the chromatic number of each graph....
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