Pamela Leutwyler

A river flows through the town of Konigsburg. 7 bridges connect the 4 land masses. While taking their Sunday stroll, the people of Konigsburg amused themselves by trying to cross each bridge EXACTLY ONCE. Eventually they became frustrated, and they sent for the mathematician Euler to explain to them why they were unable to do this. Network theory was invented.

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge

A B C D A B C D Interpret the problem with a GRAPH Each land mass is represented by a point called a vertex Each bridge is represented by a curve called an arc or edge This is called a NETWORK

A B C D The question: “Can you cross each bridge exactly once?” Becomes: “Can you trace every arc (edge) exactly once?” That is: “ Can you draw this without lifting your pencil and without retracing any arc?”

definition: A network is said to be TRAVERSABLE if you can trace each arc exactly once without lifting your pencil.

definition: A network is said to be TRAVERSABLE if you can trace each arc exactly once without lifting your pencil. This network is traversable. The Konigsburg network is not traversable. How do we know this?

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex. This vertex is even

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex. This vertex is odd

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex.

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex. If you start in an odd vertex, you will end outside of the odd vertex. out in out start end

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex. If you start in an odd vertex, you will end outside of the odd vertex. start end and if you start outside an odd vertex, you will end in the odd vertex. in out in out in

Every vertex in a network can be classified as either even or odd depending on the number of arc endings that meet in the vertex. If you start in an odd vertex, you will end outside of the odd vertex. start end and if you start outside an odd vertex, you will end in the odd vertex. A TRAVERSABLE NETWORK has AT MOST 2 ODD VERTICES. To traverse the network, you must start in one odd vertex and end in the other.

A B C D A B C D The Konigsburg bridge network is NOT traversable because there are 4 odd vertices.

If a network has all even vertices, then it is traversable. To traverse the network, you can start at any vertex and you will end where you started.