6.1 HAMILTON CIRCUIT AND PATH WORKSHEET
Use extra paper as needed.
For each of the following graphs:
1) Find ALL Hamilton Circuits starting from vertex A.
Hint: Mirror images (reverse) counts as a different circuit
2) Are there any edges that must always be used in the Hamilton Circuit?
3) Find a Hamilton Path from vertex C to E.
a. If it’s not possible, give an explanation.
GRAPH 1
GRAPH 2
A
A
B
E
D
GRAPH 3
C
A
B
E
D
C
B
E
D
C
DOES A HAMILTON PATH AND/OR CIRCUIT EXIST IN GENERAL?
For each of the following graphs: Use any vertex to START and/or END
1) Find a Hamilton Path. If it does not exist, then give a brief explanation.
2) Find a Hamilton Circuit. If it does not exist, then give a brief explanation.
A
GRAPH 1
B
C
D
GRAPH 2
A
B
C
E
F
G
H
D
F
GRAPH 5
A
GRAPH 4
A
B
GRAPH 3
B
A
E
B
F
D
C
D
C
C
E
F
E
D
E
GRAPH 6
B
C
A
GRAPH 7
A
D
C
B
D
G
F
E
F
E
H
G
6.1 HAMILTON CIRCUIT AND PATH WORKSHEET SOLUTIONS
For each of the following graphs:
4) Find all Hamilton Circuits that Start and End from A.
a. If it’s not possible, give an explanation.
5) Are there any edges that must always be used in the Hamilton Circuit?
6) Find a Hamilton Path from C to E.
a. If it’s not possible, give an explanation.
GRAPH 1
GRAPH 2
A
A
B
E
1
D
GRAPH 3
C
A
B
E
D
C
B
E
D
C
1) A, B, E, D, C, A
1) A, B, C, D, E, A
1) A, B, C, D, E, A
2) A, C, D, E, B, A
2) A, B, E, D, C, A
2) A, B, D, C, E, A
Note: B and C have
3) A, C, D, E, B, A
3) A, E, D, C, B, A
degree two so they
4) A, C, B, D, E, A
4) A, E, C, D, B, A
must use those two
5) A, D, C, B, E, A
edges
6) A, D, E, B, C, A
7) A, E, B, C, D, A
8) A, E, D, C, B, A
2
AB, BE, CD, CA
NONE
AE, AB
3
C, D, A, B, E
C, D, A, B, E
C, D, B, A, E
C, B, A, D, E
For each of the following graphs:
3) Find a Hamilton Path.
a. If it does not exist, then give a brief explanation.
4) Find a Hamilton Circuit.
a. If it does not exist, then give a brief explanation.
GRAPH 1
GRAPH 3
GRAPH 2
1) path exists
1) Path exists
1) path exists
2) No circuit because a bridge exists 2) No circuit because a bridge exists 2) circuit exists
GRAPH 5
GRAPH 4
1) Not possible
2) Not possible
4 degree 2 vertices
all share 2 adjacent
vertices
1) Possible
2) Possible
GRAPH 6
GRAPH 7
1) Possible
1) Possible
2) Not Possible
2) Possible
left degree 2 vertices
share adjacent vertex