222a6s - MATH 222 FALL 2011, Assignment Six (Answer Key) 1....

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MATH 222 FALL 2011, Assignment Six (Answer Key) 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. It follows from the Handshaking Lemma that no graph can have exactly one vertex of odd degree. Thus the two vertices of odd degree in G must be in the same (connected) component, which implies that they are joined by a path in G . 2. Find a maximum matching in the graph below. Prove that it is a maximum matching by exhibiting a minimum vertex cover. 1 2 7 3 5 9 10 8 6 4 Let M = { 14 , 36 , 58 , 710 } and C = { 3 , 7 , 4 , 8 } . Then M is a matching and C is a vertex cover. Since | M | = 4 = | C | , M is a maximum matching and C is a minimum vertex cover. 3. Let G = ( V,E ) be a simple graph with | V | ≥ 11. Prove either G or the complement G must be nonplanar. (Hint: Consider the number of edges in G and in G .) Denote n = | V | . Suppose that both G and G are planar. Then
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222a6s - MATH 222 FALL 2011, Assignment Six (Answer Key) 1....

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