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Unformatted text preview: AMS 301 Sample Exam 1 Solution Ning SUN July 27, 2009 1 True or False. 1. True. Let k be the degree of all vertices in G , and let n be the number of vertices. The degree of vertex i in G plus the degree of vertex i in ¯ G is equal to n 1 , the degree in a complete graph K n . Thus, each vertex of the complement graph must have degree n 1 k . 2. True. There cannot exist K 5 or K 3 , 3 con gurations since the graph has only 2 vertices of degree n and all other vertices have degree 2. This can also be seen by giving a planar depiction. 1. True. By Euler's Theorem, if the graph is connected and every vertex has even degree (4 is even), then an Euler cycle exists. 2 Isomorphic? The graphs are NOT isomorphic. • The graphs have the same number of vertices, 8, and all have degree 3. • The left one is bipartite (see the following bipartite depiction), but the right one is not, as it has several odd circuits, such as 123451....
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 Spring '08
 ARKIN
 Graph Theory, vertices, Bipartite graph, B. Use

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