Understanding Euler Paths and Circuits in Complete Graphs

The video tutorial explains Euler paths and circuits in graph theory, focusing on complete graphs K sub n. It defines Euler paths as walks using every edge once, existing if at most two vertices have odd degrees. Euler circuits are walks starting and ending at the same vertex, existing if all vertices have even degrees. The tutorial examines vertex degrees in complete graphs and identifies conditions for Euler paths and circuits, highlighting specific cases for different values of n.

What is an Euler path in a graph?

• A path that starts and ends at the same vertex

• A path that uses every vertex exactly twice

• A path that visits every vertex exactly once

• A path that uses every edge exactly once

Under what condition does a graph have an Euler circuit?

• All vertices have even degree

• All vertices have odd degree

• There are exactly two vertices with odd degree

• There are no vertices with even degree

What is the degree of each vertex in the complete graph K3?

• 2

• 1

• 3

• 4

In the complete graph K4, what is the degree of each vertex?

• 3

• 2

• 5

• 4

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

• 5

• 6

• 4

• 3

For which value of n does the complete graph K_n have an Euler path?

• n equals 1

• n is even and greater than 2

• n equals 3

• n is odd and greater than 1

Why does K2 have an Euler path?

• It has all vertices with even degree

• It has exactly two vertices with odd degree

• It has no vertices with odd degree

• It has more than two vertices with odd degree

For which values of n does the complete graph K_n have an Euler circuit?

• n is even

• n is odd and greater than 1

• n equals 1

• n equals 2

Why is K1 excluded from having an Euler circuit?

• It has no vertices

• It has no edges

• It has all vertices with odd degree

• It has all vertices with even degree

What is the condition for a complete graph K_n to have an Euler circuit when n is odd?

• There are exactly two vertices with odd degree

• All vertices have even degree

• There are no vertices with even degree

• All vertices have odd degree