Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

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
See similar textbooks
icon
Related questions
Question

Please provide a full solution, handwritten. 
Thanks 

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.
Expert Solution
steps

Step by step

Solved in 4 steps with 22 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS