hw8 - O 2 . (b) [3 marks] Draw a graph with the same number...

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ECE 103 Discrete Math for Engineers University of Waterloo Spring 2010 Instructors: Koray Karabina, Ashwin Nayak Homework 8, due July 19, 2010 The Frst four questions pertain to the following class of graph. The odd graph O n is the graph whose vertices are the n -subsets of { 1 , 2 , . . . , 2 n, 2 n + 1 } . Two such sets are adjacent (i.e., form an edge) if and only if they are disjoint. Question 1. (a) [2 marks] Draw O 1 , and O 2 . In the drawing for O 2 , draw the vertices, but only the edges incident on vertices { 1 , 2 } , { 2 , 4 } and { 4 , 5 } . (b) [3 marks] How many vertices and edges does O n have? Question 2. [5 marks] Prove that there is no cycle of length 3 or 4 in O 2 . Question 3. (a) [2 marks] Describe an isomorphism between O 2 and the Peterson graph drawn below, by labelling the vertices of the Peterson graph by vertices of
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Unformatted text preview: O 2 . (b) [3 marks] Draw a graph with the same number of vertices and edges as in O 2 , and in which the degree of every vertex is 3, that is not isomorphic to O 2 . Explain why it is not isomorphic. Question 4. (a) [2 marks] Describe a path in the graph O n between the vertices { 1 , 2 , . . . , n-1 , n } and { 1 , 2 , 3 , . . . , n-1 , n + 1 } . (b) [3 marks] Use the idea in part (a) to show that there is a path from the vertex { 1 , 2 , . . . , n-1 , n } to the vertex { 1 , 2 , 3 , . . . , n-2 , n + 1 , n + 2 } in the graph O n . Question 5. [5 marks] Describe a cycle in the grid graph G n , for any even n that contains all the n 2 vertices. Prove that no cycle in G 3 contains all its vertices. 1...
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This note was uploaded on 09/28/2011 for the course ECE 103 taught by Professor Nayak during the Spring '11 term at Waterloo.

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