APM 463/563—Fall 2011 Assignment #1 Due date: Thursday, Sep 22, 2011 100% for graduate students: 38 pts. 100% for undergraduate students: 36 pts. 1. (6 pts.) Prove that the number of edges in a simple bipartite graph with n ≥ 2 vertices is at most n 2 / 4. Is this result sharp? 2. (6 pts.) Let G be a simple graph on n ≥ 3 vertices such that deg G ( v ) ≥ n 2 for every vertex v ∈ V ( G ). Prove that deleting any vertex of G results in a connected graph (i.e., G is 2-connected). 3. (6 pts.) Let G be a simple graph with minimum degree δ ( G ), where δ ( G ) ≥ 2. Prove that G contains a cycle of length at least δ ( G ) + 1. Is this result sharp? 4. (6 pts.) Prove that in a connected graph every two paths of maximum length must have a common vertex. 5. (6 pts.) Prove that if K n decomposes into triangles (i.e., each edge of K n appears in exactly one of the triangles), then n-1 or n-3 is divisible by 6. Find a decomposition to triangles of K 7 and K 9 . 6. (6 pts.) Show that a loopless graph
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