T7 - mum number of edges G can have? Problem 5: You are...

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Problem 1: Suppose G is a tree with no vertices of degree 2. Prove that at least half of the vertices in G are leaves (i.e. have degree 1). Problem 2: Let G be a connected graph. Suppose the algorithm of growing a breadth first search tree produces the same tree no matter which vertex in G was chosen as the root. Prove that G is a tree. Problem 3: Let G be a graph with p vertices, q edges and minimum vertex degree k . a) Show that the number of triangles (cycles of length 3) is at least q (2 k - p ) 3 . (In particular, if p 2 < k , then the graph contains a triangle.) b) Find a graph which achieves this bound for every p . Problem 4: Suppose G is a bipartite graph on p vertices. What is the maxi-
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Unformatted text preview: mum number of edges G can have? Problem 5: You are given a standard 8 by 8 chessboard where each square has length 1. Is it possible to place 2 by 1 paper rectangles on the board so that all of the squares are covered except for two opposite corners? (Rectangles must be placed so they cover exactly two squares of the board and do not overlap) Demonstrate a valid tiling or prove none exists. Problem 6: a) Prove that every graph on 6 vertices either contains a 3-cycle or a set of 3 vertices that are pairwise non-adjacent; b) Show that the above statement is not true for graphs on 5 vertices. 1...
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