INTRODUCTION TO GRAPH THEORY WORKSHEET
The Seven Bridges of Konigsberg
You and your group members are all students of mathematics in Konigsberg, and
the year is 1735. Your city has a river that runs through it, and like many of the
inhabitants of your city, you all try to find a route through the city (pictured below)
that crosses each of the city’s bridges exactly once. Unlike most of the cities
inhabitants, however, you are all mathematicians, and are trying to understand the
question mathematically. Once more, you’ve all heard that the famous Swiss
mathematician Leonhard Euler will be arriving in town to speak at your school in
the next few days. Before he arrives you’ve all agreed to solve the ”Seven Bridges of
Konigsberg question,” to find a route across all seven bridges, or to explain why
such a route cannot exist.
Directions: First, to get familiar with the problem, try to find a path through the city
of Konigsberg that crosses each bridge only once. Do you think it can be done? Note,
the river cuts the city in half, so one cannot travel “outside” of the picture to get from
the bottom half to the top half.
INTRODUCTION TO GRAPH THEORY WORKSHEET
So, has your group come to a conclusion? You’ve probably had a hard time finding a
route through across all seven bridges. If you think that such a path cannot exist, we
need to understand why it cannot. That is, we need to prove that no solution exists.
Graph Theory Definitions
Define the following graph theory terms using complete sentences.
Vertex (Vertices):
Edges:
Graph:
Path:
Circuit:
Eulerian Path and Eulerian Circuit:
Now that we’ve learned some graph theory we can re-envision the shapes from the
beginning of class as graphs. Note, we could have added more (or less) vertices than
we have, but the following graphs should suffice.
INTRODUCTION TO GRAPH THEORY WORKSHEET
We should have seen that shapes 1, 3, 6, and 7 can be drawn without lifting our
pencil, and while 2, 4, 5, and 8 cannot. Translated into graph theory, we will say the
first group of graphs contains either an Eulerian path or an Eulerian circuit and the
second do not.
3. Your Challenge
Your group of students must decide what graph criteria are for the existence of
Eulerian paths and Eulerian circuits. Make a table using data from the above 8
graphs. Record how many vertices they have, how many edges, and the degrees of
those vertices, and categorize the graphs according to those with Eulerian paths,
those with Eulerian circuits, and those without either. Can you see a pattern
between the graphs with Eulerian paths and circuits, and the graphs without them?
Your goal will be to come up with exact requirements for a graph to have an
Eulerian path or an Eulerian Circuit. One we have these requirements, we can look
at the graph for the seven bridges problem and decided if a path or circuit over the
bridges exists or not.
Your group must do a complete write-up of the following:
1) Include your table. This should be very neat, organized, labeled and clear.
2) Include a few paragraphs on how you determined the requirements for a
Eulerian path/circuit and how you can tell by looking at a graph if it will,
or will not create a Eulerian path/ circuit.
3) Included in the paragraphs should be your final thoughts on if the
Koningberg bridge problem has a solution.
4) Include at least 5 graphs that are Eulerian paths, 5 graphs that are
Eulerian circuits and at least 5 graphs that are neither. Your Eulerican
paths/circuits should required at least 8 “moves” to complete them
(HAVE FUN WITH IT!).