Hamiltonian Circuits and
Paths
Section 4.5
Sir William Rowan Hamilton
► Irish
Mathematician
► Knighted at the age of 30
► Hamiltonian Mechanics are used to
determine orbital trajectories of satellites.
► Came up with a game…
Icosian Game
► Vertices
were meant to represent major
cities in Europe.
► Goal is to visit all 20 cities.
► Let’s PLAY!!!!!!
Street Example
► Remember
when we had the street network
and we wanted to inspect each street
exactly once? Well now we need to inspect
the fire hydrants (they are located at the
intersections. What are the intersections on
the graph?
► Can we start at the garage and stop at each
intersection exactly once and return to the
garage?
The street network
a
d
b
j
f
i
h
G
Hamiltonian Paths and Circuits
► Hamiltonian
path: You cross each vertex
exactly once and start and end at different
places.
► Hamiltonian circuit: You cross each vertex
exactly once and start and end at the same
place.
Find the Hamiltonian Circuit
Find the Hamiltonian Circuit
Find the Hamiltonian Circuit
So is there an easy rule????
► There
is not an easy rule, but there is a
vague one.
 If a connected graph has n vertices, where n >
2 and each vertex has a degree of at least n/2,
then the graph has a Hamiltonian circuit.
Does it work?
We are guaranteed to have a Hamiltonian Circuit.
Downfall: That doesn’t mean we now know how to find it.
Does it work?
We are not guaranteed a Hamilton circuit, but there is one.
Downfall: We don’t know if there is a Hamiltonian circuit.
Find the Hamiltonian Circuit
We are not guaranteed that there is a Hamiltonian circuit.
We can’t tell by the theorem that there is not one. We had
to use trial and error.
State which graphs are guaranteed
to have a Hamiltonian circuit
Let’s do another online quiz…
► http://primes.utm.edu/cgi-
bin/caldwell/tutor/graph/euler
Tournaments
► By
using directed edges, we can indicate
winners and losers in a tournament.
► An arrow from A to B means that A beats B.
► We use Hamiltonian circuits with digraphs to
determine who wins a tournament.
Example
► List
the winners of each game.
A
B
C
Why tournaments are cool!
► You
can give a ranking for each team if
there is exactly one Hamilton path.
► The winners are listed in order of the
vertices of your Hamiltonian path.
Example
► Draw
a digraph for the following school soccer
team round-robin tournament.
► Find the Hamiltonian path and determine the
ranking.
Game
AB
AC
AD
BC
BD
CD
Winner
B
A
D
B
D
D
Example
► Draw
a tournament with five players, in
which player A defeats everyone, B defeats
everyone but A, C is defeated by everyone,
and D defeats E.
Find all the directed Hamiltonian
paths for the following tournaments.
If there is only one, rank the teams.
A
D
B
C
A
D
B
C
Homework
p. 205-206 (5, 8)