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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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