hw7 - Consider a graph G on n vertices that has no cycle of...

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Computer Science 340 Reasoning about Computation Homework 7 Due at the beginning of class on Wednesday, November 14, 2007 Problem 1 Let G be a graph and u and v be two vertices in G . If u and v are connected by a walk in G , then show that G contains a path from u to v . Problem 2 In class, we defined a tree to be a connected graph with no cycles. Show that a graph G is a tree if and only if it contains no cycles, but adding any new edge creates a cycle. Problem 3 A vertex of degree 1 in a graph is called a leaf . Prove that if a tree has a node of degree d then it has at least d leaves. Problem 4 Consider a graph G ( V, E ) with costs c ij for edges ( i, j ) E . Suppose T is a spanning tree in G with the following property: For every edge e T , the cycle formed by adding e to T has no edge of cost strictly greater than c e . Prove that T is a minimum spanning tree in G . Problem 5 The girth of a graph is the length of the smallest cycle in a graph. The goal of this exercise is to show an upper bound on the number of edges of a graph in terms of its girth.
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Unformatted text preview: Consider a graph G on n vertices that has no cycle of length ≤ 2 k . Let m be the number of edges in the graph. 0. What is the average degree in G ? 1. Prove that there exists a subgraph H of G with minimum degree m/n . (Hint: Think about vertices that have degree less than m/n .) 2. Let v be a vertex in H . Consider the subgraph of H induced by vertices at distance at most k from v . Prove that this subgraph contains at least ( m/n-1) k distinct vertices. Note: The subgraph of G induced by a subset S of vertices is simply the graph G restricted to the vertices in the subset S . The vertex set of this subgraph is S ; the edges of this subgraph are all edges of G with both end points in S . In other words, all vertices outside S (and edges incident on them) are deleted. 3. Prove that m ≤ n 1+1 /k + n ....
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