**Combinatorial and Discrete Geometry: Sheet D**

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 G below:

a. How many vertices and edges does G have?

b. Write down the degree sequence of G.

c. Write down a walk that starts at x.

d. Write down an open walk that starts at x and ends at y.

e. Write down a trail that starts at x and ends at n.

f. Write down a trail that starts at x, includes m and ends at z.

g. Write down a path that starts at x and ends at n.

h. Write down a circuit that is not a cycle.

2. (Refer to graph G above.) Definition: The length of a walk is the number of edges in the walk. In particular, the length of a cycle is the number of edges in the cycle.

a. Can you find a walk of length 14 from from x to y?

b. Can you find a trail of length 14 from x to y?

c. Can you find a path of length 14 from x to y?

d. What is the length of the longest open path you can find in G?

e. What is the length of the longest cycle you can find in G?

f. Can you find a cycle that starts at x and passes through m of length 6?

g. Can you find a cycle that starts at x and passes through n of length 7?

h. Can you find a circuit of length 6 that is not a cycle?

i. Can you find a cycle of length 6 that does not include vertex r?

j. Can you find a cycle of length 6 that does include vertex r?

k. Can you find a walk that starts at vertex m and ends at vertex z and repeats the edge sw exactly 3 times?

l. Can you find a walk that starts at vertex m and ends at vertex z and repeats the edge sw exactly 4 times?

m. Can you find an open trail of length 7 that repeats exactly some vertex exactly 2 times?

n. Can you find a path of length exactly 6 which starts at r and ends at x?