Euler Paths and Euler Circuits Graph Theory Euler Paths and Euler Circuits

Euler Paths & Circuits Euler Paths and Euler Circuits T. Serino Euler Paths and Euler Circuits  (Euler is pronounced the same as Oiler) An Euler path is a path that passes through each edge of a graph exactly one time. D, B, A, C, B, E, C, D, E

Euler Paths & Circuits T. Serino An Euler circuit is a circuit that passes through each edge of a graph exactly one time. D, E, B, C, A, B, D, C, E, F, D

Euler Paths & Circuits T. Serino Although we could find Euler Paths and Euler Circuits simply by trail and error, this becomes tedious and with larger graphs it can become very difficult. To determine if an Euler Path or an Euler Circuit exists, we use Euler’s Theorem.

Euler Paths & Circuits Euler’s Theorem T. Serino Euler’s Theorem For a connected graph, the following statements are true A graph with no odd vertices (all even vertices) has at least one Euler path, which is also an Euler circuit. An Euler circuit can be started at any vertex and it will end at the same vertex. A graph with exactly two odd vertices has at least one Euler path but no Euler circuits. Each Euler path must begin at one of the two odd vertices, and it will end at the other odd vertex. A graph with more than two odd vertices has neither and Euler path nor an Euler circuit.

Euler Paths & Circuits Euler's Theorem (summary) 1. No Odd vertices: T. Serino Euler's Theorem (summary) 1. No Odd vertices:  Euler Path and Euler Circuit 2. Exactly two Odd vertices:  Euler Path, but No Euler Circuit 3. More than two Odd vertices:  No Euler Path and No Euler Circuit

Euler Paths & Circuits An Euler circuit exists, T. Serino Use Euler's Theorem to determine if an Euler Path or Euler Circuit exists: 2 A C B D F E Count the number of odd vertices. Determine the degree of each vertex. 4 4 This graph has NO Odd vertices. (all even vertices) 2 2 4 1. No Odd vertices:  Euler Path and Euler Circuit An Euler circuit exists, which is also an Euler path.

Euler Paths & Circuits An Euler Path only. T. Serino Use Euler's Theorem to determine if an Euler Path or Euler Circuit exists: A C B D F E 2 2 Count the number of odd vertices. Determine the degree of each vertex. 3 3 This graph has Exactly two Odd vertices. 2 2 An Euler Path only. 2. Exactly two Odd vertices:  Euler Path, but No Euler Circuit Notice the path shown above started at C and ended at D

Euler Paths & Circuits Neither exist. T. Serino Use Euler's Theorem to determine if an Euler Path or Euler Circuit exists: Count the number of odd vertices. 3 A C B D E Determine the degree of each vertex. ? 4 This graph has More than two Odd vertices. 3 3 ? 3 3. More than two Odd vertices:  No Euler Path and No Euler Circuit Neither exist.

Euler Paths & Circuits Solving the Konigsberg Bridge Problem: T. Serino Solving the Konigsberg Bridge Problem: Could a walk be taken through Konigsberg during which each bridge is crossed exactly one time? Remember: Locations are vertices and connections between those locations are edges. A Draw a vertex-edge graph to represent the problem. B C D

Euler Paths & Circuits T. Serino If we are trying to take a walk and cross each bridge exactly one time, an Euler path or an Euler circuit will have to be present in the graph. There are more than two odd vertices in the graph. Degree 3 A D C B Therefore, you cannot cross each bridge exactly once. This means that there is no Euler path and no Euler Circuit. Degree 5 Degree 3 Degree 3

Euler Paths & Circuits (Modified) Konigsberg Bridge Problem: T. Serino (Modified) Konigsberg Bridge Problem: If Konigsberg added some bridges to their city, could a walk be taken through which each bridge is crossed exactly one time? A B C D

Euler Paths & Circuits Modified Konigsberg Bridge Problem: T. Serino Modified Konigsberg Bridge Problem: If Konigsberg added some bridges to their city, could a walk be taken through which each bridge is crossed exactly one time? There are exactly two odd vertices in the graph. Degree 4 A Therefore, you can cross each bridge exactly once. Degree 5 This means that there is an Euler path, but no Euler circuit. B C Degree 5 D Degree 4

Euler Paths & Circuits T. Serino Fleury's Algorithm: (How to find the Euler Path or circuit) 1. Does an Euler path or Euler circuit exist? 2. Start anywhere for a circuit. The circuit will end where you started. Start at an odd vertex for a path. The path will end at the other odd vertex. 3. Start tracing edges (number them as you go) 4. Avoid dead ends. Try not to get trapped.

Euler Paths & Circuits T. Serino Recall the museum problem: Is it possible for someone to move through the museum by going through each doorway (or using each edge) of the museum exactly once? There are exactly two odd vertices in the graph. Degree 3 C F H Condor Gallery French Hamersma Jessee Muhundan Paduchowski So, yes, it is possible. (Although, the starting and ending points will be different.) Degree 3 This means that there is an Euler path, but no Euler circuit. J M P Outside

Euler Paths & Circuits To find the Euler path, start tracing edges. T. Serino To find the Euler path, start tracing edges. Because it is difficult to remember the path we traced, we should number each edge as we trace it. Because there is only an Euler path, we must start at one of the odd vertices. If we start at F, we will end at P. C F H J M P Outside 1 8 Now it is easy to follow the path we found. 2 7 9 3 4 This is one possible solution. Can you write the path? 6 5 One possible solution path is: FCJMPOMFHP

Euler Paths & Circuits T. Serino Intersections are locations, and streets are connections between those locations

Euler Paths & Circuits Try this. T. Serino Try this. The Country Oaks Neighborhood Association is planning to organize a crime stopper group in which residents take turns walking through the neighborhood with cell phones to report any suspicious activity to police. a)  Can the residents start at one intersection (vertex) and walk  each street block (edge) exactly once and return to the  intersection where they started? b) If yes, determine a circuit that could be followed.

Euler Paths & Circuits T. Serino Draw this graph in your notes and try the problem. You are looking for an Euler circuit, so you can start tracing at any point.

athematical M D ecision aking