Travel Every Edge 3. Example (Travel Every Edge 3) In the graphs above, can you find a circuit that hits every edge without repeating any edges?
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1 Travel Every Edge 3 Example (Travel Every Edge 3) In the graphs above, can you find a circuit that hits every edge without repeating any edges? Find the degrees of the vertices of the graphs above! Make a conjecture (or guess) about how to tell when you can get circuits to travel over every edge.
2 Euler Circuits 1 Example (Euler Circuits 1) Find the degrees of the vertices of the graphs above! Which of the graphs has an Euler Circuit? Where can your Euler Circuit begin and end?
3 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
4 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
5 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
6 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
7 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
8 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
9 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
10 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
11 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
12 Semi-Eulerian Graphs Definition (Semi-Eulerian Graphs) A route through a graph that covers every edge EXACTLY once with an ending vertex that is di erent than the starting vertex is called an Euler Path (also called an Euler Trail). A graph is said to be Semi-Eulerian if there is an Euler Path. An Euler Path is a little di erent. We have to start and end at di erent vertices Example (Euler Path)
13 Travel Every Edge 5 Example (Travel Every Edge 5) Can you find a path (not a round trip!) that hits every edge without repeating any edges? What are the degrees of the vertices of this graph?
14 Travel Every Edge 6 Example (Travel Every Edge 6) Can you find a path (not a round trip!) that hits every edge without repeating any edges? What are the degrees of the vertices of this graph?
15 Travel Every Edge 7 Example (Travel Every Edge 7) In the graphs above, can you find a path that hits every edge without repeating any edges? Find the degrees of the vertices of the graphs above! Make a conjecture (or guess) about how to tell when you can get path to travel over every edge.
16 Euler Paths 1 Example (Euler Paths 1) Find the degrees of the vertices of the graphs above! Which of the graphs has an Euler Path? Where must your Euler Path(s) begin and end?
17 Euler Path 2 Example (Euler Path 2) Find the degrees of the vertices of the graphs above! Which of the graphs has an Euler Path? Where must your Euler Path(s) begin and end?
18 Euler s Theorems 1: Euler Circuits We are ready for our first BIG Theorem in Graph Theory. These are the results that started Graph Theory about 300 years ago! Theorem (Euler Circuit Theorem) A connected graph has an Euler Circuit exactly when every vertex has even degree. Notice all even degrees in an example from before. Example (Degrees and Euler Circuits)
19 Euler s Theorems 2: Euler Paths Theorem (Euler Path Theorem) A connected graph contains an Euler Path exactly when all but two vertices have even degree. The Euler Path will begin and end at the vertices with odd degree. Notice all but two even degrees in an example from before. Example (Degrees and Euler Paths) WARNING: Euler s Theorems only tell us when Euler Circuits or Euler Paths exist. They do not tell us how to find them.
20 Travel Every Edge 4 Example (Travel Every Edge 4) Mtn. & Trees Mtn. Mtns. Plains Bear Mtns. & River Riv. & Trees River What are the degrees of the vertices of this graph? Do you think you can you find a circuit that hits every edge without repeating any edges?
21 Travel Every Edge 8 Example (Travel Every Edge 8) Road Rd & Mtns N. Plains River Lake Forest S. Plains John Castle Trees What are the degrees of the vertices of this graph? Do you think you can you find a path that hits every edge without repeating any edges?
22 Euler s House Euler likes to think about his house, whose blueprint is below: Can Euler travel through every doorway exactly once to make an Euler Circuit? Can Euler travel through every doorway exactly once to make an Euler Path?
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