A7 - E from G so that each component of the remaining graph...

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MATH 239 Spring 2010 Assignment 7 Due: July 9, 2010, 11:45AM EDT 1. (a) (3 marks) For p 3 , prove that a connected graph with p vertices and p edges contains exactly one cycle. (b) (3 marks) Determine the set of all graphs G with at least 3 vertices which have the property that G - e is a tree for each edge e in G . (In justifying your answer, you need to say why these are the only graphs with this property.) 2. (6 marks) Suppose that T is a tree containing only odd-degree vertices where each ver- tex has degree at most 7, and T has exactly 10 leaves. (a) What is the maximum number of vertices that T can have? Draw an example of T with this many vertices. (b) What is the minimum number of vertices that T must have? Draw an example of T with this many vertices. 3. (5 marks) Let G be a graph whose vertices have degrees either 1 or 3. Let S be the set of vertices of degree 1. Suppose that we can remove a set of edges
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Unformatted text preview: E from G so that each component of the remaining graph is a tree containing exactly one vertex in S . Prove that |E| = 1 2 | V ( G ) | . 4. (8 marks) Consider the following graph G . 1 2 3 4 5 6 7 8 9 10 11 12 13 (a) Construct a breadth-ﬁrst search tree for G , taking the vertex labelled 1 as root. When considering the vertices adjacent to the vertex being examined, take them in increasing order of their labels. (b) Determine the set of vertices whose distance to vertex 1 is two. (c) Determine whether or not G is bipartite, and prove your assertion. 5. (5 marks) Suppose that a connected graph G has a breadth-ﬁrst search tree T for which every non-tree edge joins vertices at equal levels. Prove that every cycle of G contains an even number of tree edges....
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This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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