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Unformatted text preview: and tetrahedron is dual to itself. b) The wheel is dual to itself (i.e., the dual of W n is isomorphic to W n ). 11.4.28: Vertices of a planar graph correspond to the faces of its dual, hence if G is isomorphic to its dual, then f = n . By Eulers formula, e = n + f2 = 2 n2. 3. 11.6.7: a) 2 b) 2 if n is even, 3 if n is odd c) 2, 3, 2 and 3 d) 2 (Any bipartite graph has chromatic number at most two. In the rest of the cases, it is easy to construct a 3coloring.) 4. 12.1.1: 12.1.3: a) Let e i be the number of edges and n i the number of vertices of the ith tree, for 1 i 7, e i = n i1. As  E 1  = 7 i =1 e i = 40,  V 1  = 7 i =1 n i = 47. b) Similarly, the number of components is  V 2    E 2  = 11. 12.1.15: (1) we need to exclude one edge from the cycle, giving 6 trees. (2) Similarly, we have 36 choices here....
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 Spring '09
 MarniMishna

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