To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

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Answer 1

Question

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

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Let T be a tree of order n and suppose that all vertices of T have degree 1 or degree 3. Prove that T contains exactly n-2/2 vertices of degree 3

Let G be a tree with n vertices. Find a formula for the total number of paths G and justify why it is true.

3. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers x, the graph G contains exactly x vertices of degree x, prove that two-thirds of the vertices of G have odd degree. (c) Construct a simple graph with 12 vertices satisfying the property described in part (b).

Answer 2

Question

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

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Answer 3

Question

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

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Answer 4

Question

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

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Answer 5

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

Answer 6

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

Answer 7

Chapter 12.1, Problem 27E

To determine

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

Answer 8

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

Answer 9

To determine

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

Answer 10

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Answer 11

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Answer 12

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Answer 13

Ch. 12.1 - Prob. 1TFQ Ch. 12.1 - Prob. 2TFQ Ch. 12.1 - Prob. 3TFQ Ch. 12.1 - Prob. 4TFQ Ch. 12.1 - Prob. 5TFQ Ch. 12.1 - Prob. 6TFQ Ch. 12.1 - Prob. 7TFQ Ch. 12.1 - Prob. 8TFQ Ch. 12.1 - Prob. 9TFQ Ch. 12.1 - Prob. 10TFQ

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

Solution Summary: The author explains that a tree with two vertices of degree 3 must have at least four edges. If the result isn't true, then there are two and at most three.

To prove: that a tree with vertices of degree 3 must have at least four vertices of degree1.

To prove: The result of (a) is the best possible: A tree with two vertices of degree 3 need not have five vertices of degree 1.

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Chapter 12 Solutions