Combinatorial and Discrete Geometry: Take home problem set I

(Note: You are expected to write up your solutions of these problems on your own. As long as you adhere to this whatever you do can not harm your grade for the course, only help it.)

(Graphs you draw as answers to these problems should be drawn as plane graphs whenever possible. That is, use as few accidental crossings as you can get away with.)

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

email: joeyc@cunyvm.cuny.edu
web page: http://www.york.cuny.edu/~malk

1. Given the graph below:

a. If one solves the Chinese Postman Problem for this graph how many edges must be "deadheaded?"

b. Select some minimal set of deadhead edges and find an appropriate closed walk that represents a solution to the Chinese postman problem for these deadheaded edges.

2. Given the graph G below:

a. If G has an Eulerian circuit write one down. If G has no Eulerian circuit give a reason it does not have an Eulerian circuit.

b. If G has an open Eulerian trail write one down. If G has no open Eulerian trail give a reason why.

c. Solve the Chinese postman problem for G, including giving a closed walk which achieves a minimum number of deadheads.

d. Give three of "real world problems" where finding the solution to a Chinese Postman Problem would be of use in saving money or time.

3. Draw as many graphs (no loops or multiple edges) which have the valence sequence 3, 3, 2, 2, 2, 2 as you can.

4. Draw as many graphs as you can (no loops or multiple edges) which have the valence sequence 2, 2, 2, 2, 2, 2.

5. Draw as many graphs as you can (no loops) where multiple edges are permitted which has the valence sequence 3, 3, 2, 2, 2, 2.

6. A graph with n vertices is called pancyclic if it has cycles of length 3, 4,..., n.

Determine if the graph in Problem 2. is pancyclic. If it is not pancyclic which lengths of cycles does it lack?