24.1.7. Euler's Theorem

Try these first

1. 1.

   Define what a vertex and an edge are.

   Hint

   Think of a graph as a network of points.

2. 2.

   What is the degree of a vertex?

   Hint

   Count how many edges touch a vertex.

3. 3.

   What does Euler's Theorem state?

   • The sum of the degrees is odd.
   • The number of vertices with odd degree is even.
   • Every graph must have at least one edge.

   Hint

   Think about the implications for vertex counts.

4. 4.

   True or False: A graph can have any number of odd-degree vertices.

   • True
   • False

   Hint

   Recall Euler's theorem.

5. 5.

   Given a graph with 5 vertices and 5 edges, demonstrate Euler's theorem by identifying vertices with odd and even degrees.

   Hint

   What must the total reflect about odd counts?

6. 6.

   Devise a simple graph that satisfies Euler's theorem with at least 6 vertices.

   Hint

   Sketch different configurations!