MAT 226
Homework #6
1. Suppose you and your friends decide that this
summer you will tour the north western/north
central United States by car. You will visit the
nine states shaded on the map to the right.
However, being Discrete Mathematics students,
you decide that you must cross every border
between neighboring states exactly once.
A. Can you do it?
B. If so, does it matter where (in which states)
you begin and end your road trip? Explain.
2. Consider the graph below:
a. Does it have a Hamiltonian Circuit? If so, write the vertices in order
visited. If not, explain why.
b. Does it have an Euler Path? If so, write the vertices in order visited. If not, explain why.
c. Does it have an Euler Circuit? If so, write the vertices in order visited. If not, explain why
3. Here are some complete graphs. Answer the following questions for all n, not just these drawings.
K2
K3
K4
K5
K6
a) For which n does the graph Kn contain an Euler circuit? Explain.
b) For which n does the graph Kn contain a Hamiltonian path? A Hamiltonian circuit? Explain.
4. Here are some bipartite graphs. Answer the following questions for all m and n, not just these drawings.
K2, 2
K2, 4
K3, 3
K2, 3
K2,5
K3, 5
K4, 4
K3, 4
a) For which m and n does the graph Km,n contain an Euler path? An Euler circuit? Explain.
b) For which m and n does the graph Km,n contain a Hamiltonian path? A Hamiltonian circuit? Explain.