222a6 - G or the complement G must be nonplanar(Hint...

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MATH 222 FALL 2011, Assignment Six (due Monday, Nov. 28 in class before the lecture begins) Show yourwork clearly. Illegible or disorganized solutions will receive no credit. 1. Prove that if a graph G has exactly two vertices of odd degree (and an arbitrary number of vertices of even degree), then there is a path in G joining these two vertices. [3] 2. Find a maximum matching in the graph below. Prove that it is a maximum matching by exhibiting a minimum vertex cover. [3] 1 2 7 3 5 9 10 8 6 4 3. Let G = ( V,E ) be a simple graph with | V | ≥ 11. Prove either
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Unformatted text preview: G or the complement G must be nonplanar. (Hint: Consider the number of edges in G and in G .) [3] 4. Show in each case that any graph satisfying the conditions cannot be planar. (a) A graph with 23 vertices and 65 edges. [2] (b) A graph with 23 vertices, 62 edges, and two components. [2] 5. Exhibit a graph G with a vertex v so that ( G-v ) < ( G ) and ( G-v ) < ( G ) where G-v is the graph obtained from G by deleting the vertex v and G is the complement of G . [3]...
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