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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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- Fall '07
- CharikarandChazelle
- Graph Theory, Vertex, Planar graph, G. Problem