**Combinatorial and Discrete Geometry: Sheet C**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

The goal of the exercises below is to encourage you to draw lots of graphs and check whether they are isomorphic to each other and whether they can be drawn in the plane without crossings that occur at points other than vertices (e.g. accidental crossings).

Below the following terminology will be used:

Graph: no loops or multiple edges

Multigraph: no loops, multiple edges allowed

Pseudograph: loops and multiple edges are allowed.

1. Draw a connected graph with the degree sequence given. If you are unable to draw a connected graph what about a disconnected graph. If you can not draw a graph of any kind, check to see if you can draw a multigraph or pseudograph with the given degree sequence. In each case, can you figure out how many vertices and edges the structure you are trying to draw would have to have? If you are able to draw one graph are you able to draw another which is not isomorphic to the first. Try to make your drawings without accidental crossings, that is, points where edges cross at points which are not vertices.

a. 1, 1

b. 1, 1, 1, 1

c. 1, 1, 1, 1, 1, 1

d. 1, 1, 1, 1, 1

e. 2, 1, 1

f. 2, 2, 2

g. 2, 2, 2, 2

h. 2, 2, 2, 2, 2

i. 2, 2

j. 2

2. Draw a connected graph with the degree sequence given. If you are unable to draw a connected graph what about a disconnected graph. If you can not draw a graph of any kind, check to see if you can draw a multigraph or pseudograph with the given degree sequence. In each case, can you figure out how many vertices and edges the structure you are trying to draw would have to have? If you are able to draw one graph are you able to draw another which is not isomorphic to the first. Try to make your drawings without accidental crossings, that is, points where edges cross at points which are not vertices.

a. 4, 4, 4, 4

b. 4, 4, 4, 4, 4

c. 4, 4, 4, 4, 4, 4

d. 4, 1

e. 5, 1

f. 6, 5, 4, 4, 3, 3, 3, 2

g. 6, 5, 4, 4, 3, 3, 3, 2, 2

h. 4, 4, 3, 3, 2, 2, 2, 2

i. 5, 4, 3, 2, 2, 2, 2, 1, 1

j. 5, 4, 3, 3, 2, 2, 2, 2, 1

k. 6, 4, 3, 3, 3, 3

3. Explain why if you can draw a graph with a certain degree sequence you can also draw a graph with a new degree sequence which differs from the first one only by the presence of vertices of degree 2.

4, Can you make a guess about under what conditions you can draw a pseudograph with a given degree sequence?

5. A (simple) graph is called *complete* if every vertex in the graph is joined to every other vertex in the graph. The complete graph on n vertices is denoted by K_{n}.

a. Write down the degree sequence of K_{2} and draw a graph with this degree sequence.

b. Write down the degree sequence of K_{3} and draw a graph with this degree sequence.

c. Write down the degree sequence of K_{4} and draw a graph with this degree sequence.

d. Write down the degree sequence of K_{5} and draw a graph with this degree sequence.

e. Write down the degree sequence of K_{6} and draw a graph with this degree sequence?

f. Can you eliminate any accidental crossings which might have arisen in the graphs you drew above?

6. a. Write down the degree sequence of the graph below:

b. Can you draw other graphs not isomorphic to the graph above with the same degree sequence?