Study Questions for Chapters 5,6

Study questions and advice for Chapters 5,6 Contemporary Mathematics Fall '09

Summarize the main points (at most two) for chapters 5 & 6 - how are they similar and how are they different?

What is a graph? a vertex? an edge?
In a graph, what is a circuit and what is a path (and what is the difference?)
What is an Euler circuit and an Euler path, and how are they different from circuits and paths in general?
What is the degree of a vertex, and what does it have to do with Euler paths and circuits?
How can one easily tell when a graph has an Euler circuit? an Euler path?
Bonus: Who is this Euler, anyway, and what was his influence in the development of graph theory? (and how do we say his name?)
What is the way to find Euler circuits on graphs that have them? (see section 5.6)

What are the things we mustn't do in choosing each new edge as we construct an Euler circuit?

For graphs that don't have Euler circuits, how can we find circuits that have a minimum amount of "deadhead" travel or "backtracking"?

As we add in new edges to Eulerize a graph, what are the rules for the new added edges?

Be sure that you can do the basic problems of this chapter, including telling if a specific graph has an Euler circuit and why, finding such a circuit if there is one, and how to Eulerize a specific graph.

What is a Hamilton circuit and a Hamilton path, and how are they different from Euler circuits and graphs?
There was a simple way to tell if a graph had an Euler circuit. Is there a similar simple check for Hamilton circuits? If so, what is it? (very end of 6.1).
What is a complete graph? (section 6.2)
For a complete graph with n vertices, how many Hamilton circuits are there?

...if you count each sequence of visited vertices as different?
...if you count circuits that follow the same edges (but perhaps in a different direction or starting at a different vertex) as the same?

What is a weighted graph? (p. 212)
What is a Traveling-Salesman Problem? (p. 211 - note that there need be no salesman, or even a person, involved).
If we want to find an optimal Hamilton circuit on a weighted graph, what is the only certain or guaranteed method? [Brute Force] Describe that method in two or three sentences.

Note: I should clarify - Brute Force is the only known method that is certain; people are still searching for better methods, but the problem is so difficult (of finding a fast and certain method) that it is famous.
What is the main drawback of the Brute Force method?

Sooo... if we really need an optimal Hamilton circuit for, say, a 20-city music tour, why don't we just use the one guaranteed method and be done with it? How bad could that method's disadvantages be, anyway?
Be sure that you can do the basic problems of this chapter, including using the cheapest-link, nearest-neighbor, and repeated nearest-neighbor algorithms to find approximately-optimal Hamilton circuits, and (when necessary) how to use the Brute Force method to find the surely-optimal Hamilton circuit.

Last Modified November 3, 2009.
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