Undirected graphs and networks

172 General Mathematics
Hamiltonian paths and circuits
Eulerian paths are used when we need to find a way to travel along each edge only
once. This is useful in areas such as postal or delivery routes and garbage collections.
However, there are occasions when we are interested in travelling to each vertex only
once, but it is not important that we travel along each edge.
For example, if the vertices are five tourist areas to be visited in
a town, we may be interested in visiting each area (vertex) but not
in travelling along each road. A path that passes through each
vertex once is called a Hamiltonian path (named after Sir William
Hamilton [1805–65], a Scottish mathematician). A Hamiltonian
path must pass through each vertex once, but does not have to use each edge. Usually
all of the edges are not required to draw a Hamiltonian path.
In the graph shown (above right), one Hamiltonian path is A–B–C–D–E. Another is
A–E–D–C–B. It is possible to specify a number of Hamiltonian paths from a given graph.
A Hamiltonian path that starts and finishes at the same vertex is
called a Hamiltonian circuit, in this case, one vertex is used
twice: for starting and finishing.
In the network at right, C–D–E–A–B–C is an example of a
Hamiltonian circuit.
A B
E D
B A
1. A Hamiltonian path passes through each vertex only once. It may not use all of
the edges.
2. A Hamiltonian circuit is a Hamiltonian path which starts and finishes at the
same vertex.
WORKED Example
6
Determine which of the following have a:
i Hamiltonian path
ii Hamiltonian circuit.
a B b A B C D c A
A
C
C D
F E
E
D
THINK WRITE
B E
a i 1 Specify a Hamiltonian path for the a i Begin at vertex C; travel to D–B–A–E.
given graph.
Or begin at vertex A; travel to
Note: Each vertex must be used only
once. However, each edge does not
need to be used.
There may be more than one
Hamiltonian path.
E–D–B–C.
2 Answer the question. It is possible to specify a Hamiltonian
path from the given graph. Possible
Hamiltonian paths include:
C–D–B–A–E or A–E–D–B–C.
C
D
C
E