assignment8-1 - Stratford 3 Let G be graph with girth 5 and...

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Due date: Friday, July 8 2011. Total points: 26 Math 239 Assignment 8 1. (3 points) A cubic tree is a tree whose vertices have degree either 3 or 1. Prove that a cubic tree with exactly k vertices of degree 1 has 2( k - 1) vertices. 2. Consider graph M given in Figure 1. (a) (2 points) Using a theorem that we saw in class, explain why the given spanning tree is not a breadth-first search tree rooted at Stratford . (b) (2 points) Find the breadth-first search tree rooted at Stratford . Whenever a choice arises, always select vertices according to lexicographical order. (c) (2 points) Is M bipartite? Justify your answer by referring to the breadth-first search tree that you constructed as well as to a theorem that we saw in class. Elmira Waterloo Kitchener Guelph Stratford Woodstock Cambridge Brantford London Figure 1: Graph M , representing a map of the surroundings of Kitchener , with a spanning tree T (given by thick edges), rooted at
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Unformatted text preview: Stratford . 3. Let G be graph with girth 5 and minimum degree ≥ d . (a) (2 points) For d = 3, draw such a graph with exactly d 2 + 1 vertices. (b) (3 points) Prove that any graph G with girth 5 and minimum degree ≥ d has at least d 2 + 1 vertices. 4. (3 points) Show that graph G as given in Figure 2 is planar by giving a planar embed-ding. 5. (3 points) Show that every planar bipartite graph has a vertex of degree at most 3. Math 239 Assignment 8 Due date: Friday, July 8 2011. d c b a o n m l k j i h g f e Figure 2: Graph G . 6. A planar triangulation is a connected planar embedding in which each face has degree 3. (a) (3 points) Show that every planar graph with at least 3 vertices is a spanning subgraph of a planar triangulation. (b) (3 points) Show that every planar triangulation with p vertices has 3 p-6 edges. Page 2...
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