**Combinatorial and Discrete Geometry: Sheet H**

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

Definition:

The complement of a graph G is a graph with the same vertex set as G but whose edge set consists of those edges not in G. (The edges in G and its complement together form a complete graph on the number of vertices of G.

1. For each graph below draw the complement of the graph shown

G:

H:

I:

J:

2. If G is connected and has an Eulerian circuit, must its complement also have an Eulerian circuit? (If not, give an example which shows that this statement does not hold.)

If a statement is false as shown by an example, you can try to find an additional condition which might make the statement true.

3. If G is connected must the complement of G be connected? (If not, give an example which shows that this statement does not hold.)