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Chapter 10: Q8SE (page 738)

Prove or disprove that there are always two vertices ofthe same degree in a finite multigraph having at least twovertices.

Short Answer

Expert verified

Disproven.

Step by step solution

01

Step 1:Prove or disprove that there are always two vertices of the same degree in a finite multigraph having at least two vertices.

The following is an example of a basic counterexample:

02

Result.

Hence disproven.

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Most popular questions from this chapter

Suppose that\({\bf{G}}\)is a connected multi graph with\({\bf{2k}}\)vertices of odd degree. Show that there exist\({\bf{k}}\)sub graphs that have\({\bf{G}}\)as their union, where each of these subgraphs has an Euler path and where no two of these subgraphs have an edge in common. (Hint: Add\({\bf{k}}\)edges to the graph connecting pairs of vertices of odd degree and use an Euler circuit in this larger graph.)

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Show that if \({\bf{G}}\) is a simple graph with at least 11 vertices, then either \({\bf{G}}\)or\({\bf{\bar G}}\), the complement of \({\bf{G}}\), is non-planar.
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