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Running head: GRAPH THEORY

Crossing the Bridges: Eulerian Graph Theory

By: Megan DeLaunay

History of Mathematics

Jennifer McCarthy, Annie Cox

07/25/2010

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Abstract

The Seven Bridges of Konigsberg problem, proved impossible in 1741, was the origin of

graph theory. In 1735, Leonhard Euler took interest in the problem. Konigsberg was a city in

Prussia that was separated by the Pregel River. Within the river were two more islands. The four

landmasses had seven bridges connecting them. A goal of the people of Konigsberg was to try to

make a journey across all seven bridges without recrossing a bridge. In 1741, Euler deemed this

journey impossible due to contradiction. A journey with seven bridges would need to touch eight

landmasses. However, with his formula (x+1)/2, where x is any odd number of bridges stemming

from one vertex (landmass), Euler proved that the journey would need to touch nine, not eight,

landmasses, deeming it impossible. Euler then translated this proof into a general theorem,

Euler’s Theorem, which acts as the basis of graph theory. This general theorem can then be used

to solve similar problems, such as if an Eulerian circuit path is possible over nineteen bridges in

Pittsburgh, PA.

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Background

Seven Bridges of Königsberg

Königsberg (now called Kaliningrad, Russia) was a city in Prussia along the coasts of the

Baltic Sea and the Pregel River. In the river running through the city, there were two islands;

each island and each shore of the mainland was connected using a series of seven bridges.

The seven bridges were called Blacksmith’s bridge, Connecting Bridge, Green Bridge,

Merchant’s Bridge, Wooden Bridge, High Bridge, and Honey Bridge (Paoletti). The citizens of

Königsberg were curious to see if it was possible to travel across each bridge only once in one

journey. For years, citizens tried to succeed, each time either recrossing or skipping a bridge.

(Smith p. 105)

Figure 1. Map of Konigsberg Bridge Problem (Muhammad).

Figure 2. Possible Journey. This figure shows a possible unsuccessful

pathway that could be taken. In order to cross the final bridge, at least

one bridge would have to be crossed twice (Muhammad).

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Finally, the town concluded that it was impossible to cross each bridge, but no one, not

even the greatest of mathematicians in Königsberg, knew why.

Years later, in around 1735, Leonhard Euler took interest into the problem. This problem

seemed so trivial, though, that many questioned why Euler would even take part in it. The

answer comes from its lack of an algebraic or geometric answer – the Königsberg bridge

problem was in a field all to its own. Euler ended up being the first mathematician to use graph

theory in his explanation of why it was impossible.

Introduction to Graph Theory

Graph theory began in the hands of Euler and his work with the Königsberg Bridges

Problem in 1735. Euler, at the forefront of numerous mathematical concepts at his time, was the

first to propose a solution to the Königsberg Bridges Problem.

Modern day graph theory has evolved to become a major part of mathematics used for

solving puzzle-like problems. In order to fully understand graph theory, the following terms need

to be defined:

Degree –the number of edges with that vertex as an end-point

Walk – “a way of getting from one vertex to another” and consists of a sequence

of edges, one following another

Trail – a walk in which all the edges are distinct (only appear once)

Path – a walk where no vertex appears more than once

Cycle – a closed path that returns back to the starting point

Bridge – the only edge connecting two sections of a graph

These terms are vital to understanding the rest of Euler’s proof and Eulerian graph theory

as a whole (Muhammad).

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Euler’s Proof

The paper Euler produced in 1741, roughly translating to “Learning of Problems

Pertaining to Geometric Regions” twenty-one paragraphs long and proved that the Königsberg

Bridge Problem was indeed impossible to complete. In his paper, he begins with explain the

Seven Bridges Problem and providing a sketch of a map. He then gives each bridge a variable, a-

g. Within his paper, Euler tries many different methods, but concludes each known method to be

incorrect for this problem and so begins search for a new method entirely.

For his proof, Euler drew the map as a network of vertices and arcs. Vertices I1 and I2

represented the islands, and vertices S1 and S2 represented the two shores. The arcs (or edges) in

between the vertices represented the bridges.

Euler begins by deciphering a method to label each landmass instead of bridge, using

capital letters. “For instance, AB would signify a journey that started in landmass A, and ended

in B. Furthermore, if after traveling from landmass A to B, someone decides to move to

landmass D, this would simply be denoted, ABD. Euler continues his discussion on this process

explaining that in ABDC, although there are four capital letters, only three bridges were crossed.

Euler explains that no matter how many how many bridges there are, there will be one more

S1

I1 I2

S2

Figure 3. Euler’s graph of Konigsberg (Wilson, 1972).

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letter to represent the necessary crossing. Because of this, the whole of the Königsberg Bridge

problem required seven bridges to be crossed, and therefore eight capital letters” (Paoletti).

The first problem Euler encountered occurred with the fact that there were two bridges

leading from point A to B. However, he concluded that it would not matter which bridge was

chosen, as long as the walker got from point to point.

The point of Euler’s proof was to either find the eight letter sequence that completes the

problem, or proof that is does not exist. Euler decided that the Konigsberg problem was too

difficult to figure out any rules, so he started with a simpler problem. After studying a problem

with vertex A and different odd numbers of bridges coming from it, Euler concludes the equation

(x+1)/2 to be the correct number of times a vertex must appear in a sequence in all cases where x

= an odd number of bridges.

Looking back at the Konigsberg bridge problem, Euler deduces that landmass A needs to

appear three time, and B, C, and D each need to appear twice since there are three bridges

leading to each. Therefore, 3(A) + 2(B) + 2(C) + 2(D) = 9. However, Euler has already

concluded that the problem would need a sequence of eight landmasses since there are seven

bridges. Since 8 ≠ 9, it can be said that the path would be impossible due to the contradiction.

Euler’s Theorem

Euler’s proof led to the development of Euler’s Theorem, a theorem that can be used to

help solve graphs and conclude if they are or are not “Eulerian”. A graph contains an Eulerian

circuit (therefore being labeled Eulerian) if and only if the closed graph contains a circuit in

which each side is traced only once. Graphs like the Konigsberg Bridge graph do not contain

Eulerian circuits.

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A graph is labeled semi-Eulerian if it contains a trail where each edge is only crossed

once, but does not return to the starting point.

Euler’s theorem states that:

A graph with more than two odd vertices (vertices having an odd number of edges

stemming from them) does not contain a Eulerian circuit

A graph with exactly two odd vertices is semi-Eulerian

A graph with no odd vertices contains a Eulerian circuit

Following Euler’s proof, the Fleury algorithm was established in order to provide a

method of finding an Eulerian circuit within a graph. The algorithm says:

Figure 4. This graph is Eulerian because the walk with the sequence

ABCDECGEFGBFA contains each side and returns to starting point A.

Figure 5. This graph is semi-Eulerian because a trail is contained that traces

each edge. However, starting at point A, it is impossible to end to point A and

therefore ends at point D.

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1. Apply Euler’s theorem – Will the graph be Eulerian, Semi, or neither?

2. If there are no odd vertices, start anywhere. If there are two odd vertices, start at

an odd vertex (the other odd vertex will be the endpoint)

3. Trace the edges as you move through the graph.

4. If a bridge exists, only cross it when all other edges on one side of a graph have

been traced.

Euler’s General Form

Euler modified his proof of the Konigsberg bridge problem to solve any similar situation

by setting up a chart and a six-step algorithm. The goal is to decide is there is a journey possible

in which each edge is crossed only once.

1. Denote each landmass with a capital letter

2. Count the total number of bridges, record at the top of the chart. Add one, and record

that number as well.

3. After landmasses have been labeled accordingly, list the capital letters in a column

and record the number of bridges to and from each in a new column directly to the

right.

4. Indicate the landmasses that have an even number of bridges with an asterisk (*),

Figure 6. In this figure, FD is a bridge and so can only be crossed after

EF, EG, and GF have been traced on the left side.

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5. In a third column, next to even number of bridges, record exactly half the number.

This number is the number of times this landmass will be passed through during the

journey. Leftover should be the odd numbers of bridges. Add one to the number to

make it even, and in the third column record half as was done with the even numbers.

6. Finally, add the numbers in the right-most column. This is the sum

If the sum is one less than or equal to the number of bridges plus one (recorded on the top

of the chart), then the journey is possible. However, if it is one less, the journey must start with a

landmass denoted with an asterisk. If the sum is equal, then the journey must start at a landmass

not denoted by an asterisk (Paoletti).

Examples

An example of the system above is something like the following:

Number of Bridges = 14 + 1 = 15

Region Bridges Times Must Appear

A* 8 4

B* 4 2

C* 4 2

D 3 2

E 5 3

F* 6 3

Sum = 16

Since in this example, the sum is sixteen and the number of bridges plus one (recorded at

the top) is fifteen, it is known that a journey will be possible as long as it starts at landmass D or

E. This goes back to Euler’s original theorem, in which he states that a graph with two

landmasses with an odd number of bridges will have a journey if it begins at the odd vertices

(Paoletti).

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The city if Pittsburgh, Pennsylvania has the most bridges of any city in the United States

with four hundred forty six bridges. In just the approximately five square mile area of central

Pittsburgh, including Herrs and Brunots Island, there are nineteen bridges between five

landmasses. The bridges cross the Allegheny River, Ohio River, and Monongahela River and

connect downtown Pittsburgh to its suburbs as well as Brunots and Herrs Islands (Pittsburgh). In

order to decide if there is a way to cross each of these Bridges and end back in central Pittsburgh,

Euler’s general system needs to be used.

\

In order to assess whether or not an Eulerian circuit could be found, a chart was set up

like the one below.

Number of Bridges = 19 + 1 = 20

Region Bridges Times Must Appear

A* 14 7

B* 2 2

C* 12 6

D* 2 1

E* 8 4

Sum = 20

Figure 7. Map of Pittsburgh. With bridges highlighted in purple and landmasses labeled

(Google Maps).

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Since in this case, the sum of the times each landmass must appear is equal to the original

number of bridges plus one, it can be stated that an Eulerian circuit can be found starting and

finishing at any landmass.

Conclusion

The founding of graph theory changed the pace of non-algebraic or geometric problems

in mathematics. Euler’s accomplishments led to the development of a whole new field. Graph

theory is a relatively new development that is still being researched today.

The Konigsberg Bridge Problem laid a foundation for this entire field. Without the people

of Konigsberg’s inquiries about their Sunday journeys, graph theory may not exist. Euler’s proof

allows us to delve deeper into networks and graphs that have many applications in the modern

world. For example, electricians often use networks to decide how wires need to be connected.

Euler’s proof, although extensive, allowed for the development of his theorem, Fleury’s

algorithm, and a general form for solving networks and graphs, as well as answering questions

about their possible circuits and journeys. For example, it could be deduced that the city of

Pittsburgh contains an Eulerian circuit across five different landmasses which allows for better

planning and better directions.

Although graph theory does not include much of what most people consider

“mathematics”, with almost no numbers or equations, graph theory combines aspects across

fields of mathematics.

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References

Just how many Bridges are there in Pittsburgh?. (2006, September 13). The Pittsburgh Channel,

Retrieved from http://www.thepittsburghchannel.com/news/9841603/detail.html

Muhammad, R. B. (n.d.). Graph theory. Retrieved from

http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/graphIntro.ht

Paoletti, T. (n.d.). Leonard euler's solution to the konisberg bridge problem. Retrieved from

http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2429&pf=1

Smith, S. M. . (1996). Agnesi to zeno: over 100 vignettes from the history of math. Emeryville,

CA: Key Curriculum Press.

Vignette 5: graph theory and the bridges of konisberg. (2000, August 28). Retrieved from

http://www.jcu.edu/math/vignettes/bridges.htm

Wilson, R. J. (1972). Introduction to graph theory. India : Pearson Education.