a6 - n-subsets of the set { 1 , 2 ,..., 2 n +1 } , and...

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MATH 239 Assignment 6 This assignment is due at 11:30 on Friday, November 6, 2009, in the drop box corresponding to your section. 1. Let G be a graph with p vertices, where p 1. Suppose every vertex in G has degree greater than or equal to b p 2 c . Prove that G is connected. 2. (a) Let G be a graph such that exactly two vertices of G have odd degree. Show that these two vertices are in the same component of G . (b) Let k be even. Using part (a), or otherwise, show that a k -regular graph cannot contain a bridge. 3. The odd graph O ( n ) is the graph whose vertices are the set of
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Unformatted text preview: n-subsets of the set { 1 , 2 ,..., 2 n +1 } , and where two vertices are adjacent if and only if they are disjoint. (a) Show that O (2) is isomorphic to the Petersen graph depicted in Figure 1. (b) Prove that O ( n ) is connected. Figure 1: The Petersen graph 4. Let G be a graph with at least two vertices, such that the cut induced by every proper nonempty subset of vertices of G contains exactly two elements. Determine (with proof) the number of vertices and edges in G . 1...
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