Aim: What is an Euler Path and Circuit? Do Now: Represent the following with a graph

Euler Path Euler Path – a path that travels through every edge of a graph once and only once. Each edge must be traveled and no edge can be retraced. 6 1 5 2 7 8 9 4 3 A, B, E, F, D, B, C, E, D, G

Every Euler circuit is an Euler path Euler Circuit – a circuit that travels through every edge of a graph once and only once, and must begin and end at the same vertex. 6 A, B, E, F, D, 1 5 2 7 10 8 B, C, E, D, G, A 9 4 3 Every Euler circuit is an Euler path Not every Euler path is an Euler circuit Some graphs have no Euler paths Other graphs have several Euler paths Some graphs with Euler paths have no Euler circuits

Euler Theorem Euler’s Theorem The following statements are true for connected graphs: If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other. If a graph has no odd vertices (all even vertices), it has at least one Euler circuit. An Euler circuit can start and end at any vertex. If a graph has more than two odd vertices, then it has no Euler paths and no Euler circuits.

Explain why the graph has at least one Euler path. Model Problem Explain why the graph has at least one Euler path. degree 2 D C B E A number of edges at each vertex: degree 4 degree 4 A: 2 D: 3 B: 4 E: 3 C: 4 degree 3 degree 3 Two odd vertices If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other.

Model Problem Find a Euler path. D C B E A D, C, B, E, C, A, B, D, E 6 5 2 3 7 4 1 8 D, C, B, E, C, A, B, D, E

Explain why the graph has a least one Euler circuit. Model Problem Explain why the graph has a least one Euler circuit. degree 2 degree 4 degree 4 degree 2 degree 2 degree 4 degree 4 degree 2 no odd vertices 2. If a graph has no odd vertices (all even vertices), it has at least one Euler circuit. An Euler circuit can start and end at any vertex.

Model Problem Find an Euler circuit. H, G, E, C, G, I, J, H, D, C, A, 11 10 12 9 3 13 4 8 2 14 1 5 7 6 H, G, E, C, G, I, J, H, D, C, A, B, D, F, H Euler Circuit – a circuit that travels through every edge of a graph once and only once, and must begin and end at the same vertex.

Graph Theory’s Beginnings In the early 18th century, the Pregel River in a city called Konigsberg, surround an island before splitting into two. Seven bridges crossed the river and connected four different land area. Many citizens wished to take a stroll that would lead them across each bridge and return them to the starting point without traversing the same bridge twice. Possible? They couldn’t do it. Euler proved that it was not possible.

The Theorem at Work R L A B R L A B degree 3 degree 5 3. If a graph has more than two odd vertices, then it has no Euler paths and not Euler circuits.

a) Is it possible to find a path that uses each door exactly once? Model Problem A, B, C, and D represent rooms. The outside of the house is labeled E. The openings represent doors. a) Is it possible to find a path that uses each door exactly once? degree 6 degree 3 look for a Euler path or circuit exactly 2 odd vertices If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other.

b) If possible, find such a path. Model Problem A, B, C, and D represent rooms. The outside of the house is labeled E. The openings represent doors. b) If possible, find such a path. 9 10 7 2 1 3 8 6 4 5 start at one of the odd vertices B, E, A, B, D, C, A, E, C, E, D

Fluery’s Algorithm If Euler’s Theorem indicates the existence of an Euler path or Euler circuit, one can be found using the following procedure. If the graph has exactly two odd vertices, chose one of the two odd vertices as the starting point. If the graph has no odd vertices, choose any vertex as the starting point. Number edges as you trace through the graph according to the following rules: After you traveled over an edge, change it to a dashed line. When faced with a choice of edges to trace, choose an edge that is not a bridge (an edge, which, if removed from a connected graph would leave behind a disconnected graph). Travel over an edge that is a bridge only if there is no alternative.

Model Problem no odd vertices, begin at any vertex. The graph has at least one Euler circuit. Find it using Fleury’s Algorithm. F 7 E 6 4 3 8 2 D C 1 9 5 B A no odd vertices, begin at any vertex.