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Let the degree sequence of a graph G be the sequence of length |V(G)| that contains the degrees of the vertices of G in non-increasing order. (a) For each of the following sequences, either draw a simple graph whose de- gree sequence is equal to that sequence, or explain why such a graph does not exist: (i) (4, 4, 4, 2, 2), (ii) (4, 2, 2, 1, 1), (iii) (3, 3, 3, 2, 1), (iv) (4, 3, 3, 2, 1), (v) (2, 2, 2, 1, 1). (b) Consider a simple graph with 9 vertices, such that the degree of each vertex is either 5 or 6. Prove that there are at least 5 vertices of degree 6 or at least 6 vertices of degree 5.

Let the degree sequence of a graph G be the sequence of length |V(G)| that contains the degrees of the vertices of G in non-increasing order. (a) For each of the following sequences, either draw a simple graph whose de- gree sequence is equal to that sequence, or explain why such a graph does not exist: (i) (4, 4, 4, 2, 2), (ii) (4, 2, 2, 1, 1), (iii) (3, 3, 3, 2, 1), (iv) (4, 3, 3, 2, 1), (v) (2, 2, 2, 1, 1). (b) Consider a simple graph with 9 vertices, such that the degree of each vertex is either 5 or 6. Prove that there are at least 5 vertices of degree 6 or at least 6 vertices of degree 5.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter1: Equations And Graphs
Section1.2: Graphs Of Equations In Two Variables; Circles
Problem 5E: a If a graph is symmetric with respect to the x-axis and (a,b) is on the graph, then (,) is also on...
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Let the degree sequence of a graph G be the sequence of length |V(G)| that contains the degrees of the vertices of G in non-increasing order. (a) For each of the following sequences, either draw a simple graph whose de- gree sequence is equal to that sequence, or explain why such a graph does not exist: (i) (4, 4, 4, 2, 2), (ii) (4, 2, 2, 1, 1), (iii) (3, 3, 3, 2, 1), (iv) (4, 3, 3, 2, 1), (v) (2, 2, 2, 1, 1). (b) Consider a simple graph with 9 vertices, such that the degree of each vertex is either 5 or 6. Prove that there are at least 5 vertices of degree 6 or at least 6 vertices of degree 5.
Transcribed Image Text:Let the degree sequence of a graph G be the sequence of length |V(G)| that contains the degrees of the vertices of G in non-increasing order. (a) For each of the following sequences, either draw a simple graph whose de- gree sequence is equal to that sequence, or explain why such a graph does not exist: (i) (4, 4, 4, 2, 2), (ii) (4, 2, 2, 1, 1), (iii) (3, 3, 3, 2, 1), (iv) (4, 3, 3, 2, 1), (v) (2, 2, 2, 1, 1). (b) Consider a simple graph with 9 vertices, such that the degree of each vertex is either 5 or 6. Prove that there are at least 5 vertices of degree 6 or at least 6 vertices of degree 5.
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