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Midterm-Solutions copy

Midterm-Solutions copy - MATH 137A Midterm Friday Solutions...

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Unformatted text preview: MATH 137A Midterm Friday, February 11, 2011 Solutions 1. Determine whether K 4 contains the following (give an example or a proof of non- existence). (a) A walk that is not a trail. (b) A trail that is not closed and is not a path. (c) A closed trail that is not a cycle. Solution: A labeled complete graph on four vertices is given below. • v 1 e 1 e 4 e 5 L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L • v 2 e 2 • v 4 e 6 r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r • v 3 e 3 (a) A trail is a walk with all of its edges distinct. Therefore w = ( v 1 ,e 1 ,v 2 ,e 6 ,v 4 ,e 3 ,v 3 ,e 2 ,v 2 ,e 1 ,v 1 ) is one example of a walk that is not a trail. (b) The walk w = ( v 1 ,e 5 ,v 3 ,e 2 ,v 2 ,e 6 ,v 4 ,e 3 ,v 3 ) is a trail, since no edges are repeated. It is not closed, since the initial and final vertices are not the same. It is not a path, since the vertex v 3 is repeated. (c) There is no closed trail in K 4 that is not a cycle. Such a walk would have to repeat no edge (so as to be a trail) and at the same time repeat a vertex, say u , that is not the initial/final vertex (so as to not be a cycle). Similar to the case of the Bridges of K¨ onigsberg problem, the degree of u would have to be at least four. This is because u would have to be “entered” then “exited” then “reentered” then “reexited”, all on different edges. Since the maximum degree of a vertex in K 4 is 3, no such walk exists. 1 2. Find three nonisomorphic simple graphs on six vertices where two vertices have degree 3 and four vertices have degree 2. Solution: Adding an edge to C 6 , the cycle graph on 6 vertices, gives one example....
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Midterm-Solutions copy - MATH 137A Midterm Friday Solutions...

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