Assn9Soln - and tetrahedron is dual to itself. b) The wheel...

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1. A connected graph has Euler circuit if and only if each vertex has even degree. This is the case if and only if each face of the dual graph has even size. Observe that this is the case if and only if every cycle in the dual has even length (the interior of the cycle can be decomposed to faces, and the overlaps of their boundaries decrease the total length by an even number). Finally, a graph is bipartite if and only if all of its cycles have even length. 2. 11.4.23: a) As each face has size at least k , 2 e kf , where f is the number of faces. By Euler’s formula, e + 2 - n = f 2 k e , and this simpli±es to e k k - 2 ( n - 2). b) 4 c) K 3 , 3 has 6 vertices, by the formula a planar graph with 6 vertices and minimal cycle length 4 has at most 8 edges, but K 3 , 3 has 9 edges. d) Similarly, for the Petersen graph we have n = 10 and k = 5, giving the upper bound of 40 3 < 14 edges, while the Petersen graph has 15 edges. 11.4.25: a) cube is dual to octahedron, dodecahedron to icosahedron,
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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 + f-2 = 2 n-2. 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 3-coloring.) 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 i-th tree, for 1 i 7, e i = n i-1. 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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Assn9Soln - and tetrahedron is dual to itself. b) The wheel...

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