Graph Theory Terminology
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Adjacent Vertices

Adjacency Matrix

Adjacent Edges

Degree of a Vertex (Valence)

Odd and Even Vertex

Path

Circuit

Connected Graph

Bridge

Euler Path

Euler Circuit
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Hamiltonian Path

Hamiltonian Circuit
Graph Theory Explorations
What follows are several questions designed to initiate discussion about different graph theory
concepts. While obtaining an answer these questions is part of the process, the larger goal is to
uncover concepts, algorithms, and theorems about graph theory. As you explore these questions,
try to generalize your findings. Explorations #1 - #3 deal with Euler paths and circuits while #4
and #5 relate to Hamiltonian paths and circuits.
References (including many of the images)
Tannenbaum, P. (2007). Excursions in Modern Mathematics with Mini-Excursions (Sixth ed.). Upper
Saddle River, New Jersey: Pearson Prentice Hall.
Wikipedia (topics: graph theory, Seven Bridges of Königsberg, and Hamiltonian path) on 2/11/2009
Exploration #1: Seven Bridges of Königsberg
Part A
The three “diagrams” above are equivalent according to graph theory; explain.
Part B
There is a variation of the Seven Bridges of Königsberg
in which each node is given a unique name or color.
The northern bank of the river is occupied by the Schloß,
or castle, of the Blue Prince; the southern by that of the
Red Prince. The east bank is home to the Bishop's
Kirche, or church; and on the small island in the center is
a Gasthaus, or inn.
It being customary among the townsmen, after some hours in the Gasthaus, to attempt to “walk
the bridges”, many have returned for more refreshment claiming success. However, none have
been able to repeat the feat by the light of day. “Walking the bridges” means that they have
crossed every bridge exactly one time (swimming in the river is not allowed).
BRIDGE 8: The Blue Prince, having analyzed the town's bridge system by means of graph
theory, concludes that the bridges cannot be walked. He contrives a stealthy plan to build an
eighth bridge so that he can begin in the evening at his Schloß, walk the bridges, and end at the
Gasthaus to brag of his victory. Of course, he wants the Red Prince to be unable to duplicate the
feat. Where does the Blue Prince build the eighth bridge?
BRIDGE 9: The Red Prince, infuriated by his brother's Gordian solution to the problem, wants
to build a ninth bridge, enabling him to begin at his Schloß, walk the bridges, and end at the
Gasthaus to rub dirt in his brother's face. His brother should then no longer walk the bridges
himself. Where does the Red Prince build the ninth bridge?
BRIDGE 10: The Bishop has watched this furious bridge-building with dismay. It upsets the
town's Weltanschauung and, worse, contributes to excessive drunkenness. He wants to build a
tenth bridge that allows all the inhabitants to walk the bridges and return to their own beds.
Where does the Bishop build the tenth bridge?
Exploration#2: The Bridges of Madison County
Madison County is a quaint old place, famous for its quaint old bridges. A beautiful river runs
through the county, and there are four islands (A, B, C, and D) and 11 bridges joining the islands
to both banks of the river (R and L) and one another (see picture above). A famous photographer
is hired to take pictures of each of the 11 bridges for a national magazine. The photographer
needs to drive across each bridge once for the photo shoot. Moreover, since there is a $25 toll
(the locals call it a “maintenance tax”) every time an out-of-town visitor drives across a bridge,
the photographer wants to minimize the total cost of his trip and to recross bridges only if it is
absolutely necessary. What is an optimal (cheapest) route for him to follow?
Exploration#3: On Patrol and the Postman Cometh
Part A
After a rash of burglaries, a private security guard is hired to patrol the streets of the Sunnyside
neighborhood (shown above). The security guard’s assignment is to make an exhaustive patrol,
on foot (or Segway) through the entire neighborhood. Obviously, she doesn’t want to walk any
more than what is necessary. Her starting point is the southeast corner across from the school (S
in the picture) – that’s where she parks her car. Being mathematically inclined, the security
guard has two questions she would like to know the answer to: (i) is it possible to start and end at
S, cover every block of the neighborhood, and pass through each block once?, and (ii) if some of
the blocks will have to be covered more than once, what is an optimal route that covers the entire
neighborhood? (Optimal here means “with the minimal amount of distance covered.”)
Part B
The Sunnyside post office needs to develop a route for delivering the mail in Sunnyside.
Because there are house on both side of certain streets, the postal worker will need to travel
down those roads twice (once on each side). Also, certain roads in Sunnyside only have houses
on one side of the road and some roads have no houses on them. If the delivery route must begin
and end at the post office, what is an optimal route?
Exploration#4: Platonic Solids
To the left is a 2D representation of a dodecahedron that
preserves the adjacency of the vertices. The solid line
shows that a Hamiltonian circuit exists for a
dodecahedron. Select one of the other five platonic solids
below and show that a Hamiltonian circuit exists.
The Five Platonic Solids
Tetrahedron
(v=4)
Cube
(or Hexahedron)
(v=8)
Octahedron
(v = 6)
Dodecahedron
( v = 20 )
Icosahedron
( v = 12 )
Exploration#5: A Tale of Five Cities
A
B
C
D
E
A
$185
$119
$152
$133
B
$185
$121
$150
$200
C
$119
$121
$174
$120
D
$152
$150
$174
$199
E
$133
$200
$120
$199
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Meet Willy, a traveling salesman. Willy has customers in five cities, which we will call A, B, C,
D, and E. Willy needs to schedule a sales trip that will start and end at A (that’s Willy’s
hometown) and goes through each of the other four cities. Other than starting and ending at A,
there are no restrictions as to the sequence in which he visits the other cities. The matrix shows
the cost of a one-way airline ticket between each pair of cities. Draw the graph represented by
the matrix. What is the optimal (cheapest) Hamilton circuit? How did you find your answer?