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1. Prove that a graph is bipartite if and only if it doesn't have any simple circuits of odd order. Hint. Use the fact that a graph is bipartite if and only if it is 2-colorable.

1. Prove that a graph is bipartite if and only if it doesn't have any simple circuits of odd order. Hint. Use the fact that a graph is bipartite if and only if it is 2-colorable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Prove that a graph is bipartite if and only if it doesn't have any simple circuits of odd order. Hint. Use the fact that a graph is bipartite if and only if it is 2-colorable. 2. Are the two graphs in Figure 1 isomorphic? How about the graphs in Figure 2? or in Figure 3? Prove your answers. Figure 1: The first two graphs of Question 2 Figure 2: The middle two graphs of Question 2 3. Are the graphs in Figure 4 isomorphic? Prove your answer. 4. How many cut vertices does the path P, has for n 2 1? Figure 3: The last two graphs of Question 2 E H 10 Figure 4: The two graphs of Question 3
Transcribed Image Text:1. Prove that a graph is bipartite if and only if it doesn't have any simple circuits of odd order. Hint. Use the fact that a graph is bipartite if and only if it is 2-colorable. 2. Are the two graphs in Figure 1 isomorphic? How about the graphs in Figure 2? or in Figure 3? Prove your answers. Figure 1: The first two graphs of Question 2 Figure 2: The middle two graphs of Question 2 3. Are the graphs in Figure 4 isomorphic? Prove your answer. 4. How many cut vertices does the path P, has for n 2 1? Figure 3: The last two graphs of Question 2 E H 10 Figure 4: The two graphs of Question 3
E H 10 Figure 4: The two graphs of Question 3 Page 2 5. Show that the following graphs have no cut vertices: (a) Cn for n > 3. (b) Wn for n > 3. (c) Qn for n > 1. 6. For which values of m and n does the complete bipartite graph Km,n have cut vertices?
Transcribed Image Text:E H 10 Figure 4: The two graphs of Question 3 Page 2 5. Show that the following graphs have no cut vertices: (a) Cn for n > 3. (b) Wn for n > 3. (c) Qn for n > 1. 6. For which values of m and n does the complete bipartite graph Km,n have cut vertices?
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