Understanding Euler Paths and Circuits

The video tutorial explains Euler paths and circuits, detailing the conditions for their existence in graphs. It analyzes several graphs, including K4, K5, K3,5, K2,3, C5, and P4, to determine whether they have Euler paths or circuits based on vertex degrees. The tutorial concludes with practical examples to illustrate these concepts.

What is a necessary condition for a graph to have an Euler path?

• All vertices must have odd degree.

• All vertices must have even degree.

• Exactly two vertices must have odd degree.

• At most two vertices can have odd degree.

Which of the following is true for a graph to have an Euler circuit?

• All vertices must have odd degree.

• Exactly two vertices must have odd degree.

• At least one vertex must have odd degree.

• All vertices must have even degree.

Why does the complete graph K4 not have an Euler path?

• All vertices have even degree.

• It is not connected.

• More than two vertices have odd degree.

• It has a loop.

What is the degree of each vertex in the complete graph K5?

• 3

• 4

• 5

• 6

Why does the complete bipartite graph K3,5 not have an Euler path?

• All vertices have even degree.

• It is not connected.

• All vertices have odd degree.

• It has a loop.

What is the degree of vertices in the top row of the complete bipartite graph K2,3?

• 2

• 3

• 4

• 5

What characteristic of the cycle graph C5 allows it to have an Euler circuit?

• All vertices have odd degree.

• All vertices have even degree.

• It is a complete graph.

• It is a bipartite graph.

What is the degree of each vertex in the cycle graph C5?

• 1

• 2

• 3

• 4

How many vertices with odd degree does the path graph P4 have?

• 1

• 0

• 3

• 2

Why does the path graph P4 not have an Euler circuit?

• It has more than two vertices with odd degree.

• It has a loop.

• It is not connected.

• Not all vertices have even degree.