The problem asks to determine whether an Euler path or an Euler circuit exists for each of the given graphs $G_1$, $G_2$, and $G_3$. Let's define these terms before analyzing each graph:

• Euler Path: A path in a graph that visits every edge exactly once. An Euler path exists if the graph has exactly two vertices of odd degree (or none if it is a circuit).
• Euler Circuit: A circuit (closed path) that visits every edge exactly once. An Euler circuit exists if every vertex in the graph has an even degree.

Now, let’s examine each graph:

1. Graph $G_1$:

• Vertices: A, B, C, D, E
• The degree of each vertex:
  • $\deg(A) = 2$ (connected to B, C)
  • $\deg(B) = 3$ (connected to A, C, D)
  • $\deg(C) = 3$ (connected to A, B, D)
  • $\deg(D) = 3$ (connected to B, C, E)
  • $\deg(E) = 1$ (connected to D)

Euler Path: Since there are exactly two vertices of odd degree (B, C, and D), an Euler path exists.

Euler Circuit: Since not all vertices have even degrees, there is no Euler circuit.

2. Graph $G_2$:

• Vertices: A, B, C, D, E, F
• The degree of each vertex:
  • $\deg(A) = 4$
  • $\deg(B) = 4$
  • $\deg(C) = 4$
  • $\deg(D) = 4$
  • $\deg(E) = 4$
  • $\deg(F) = 4$

Euler Path: All vertices have even degrees, so an Euler path exists.

Euler Circuit: Since all vertices have even degrees, an Euler circuit also exists.

3. Graph $G_3$:

• Vertices: A, B, C, D, E
• The degree of each vertex:
  • $\deg(A) = 4$
  • $\deg(B) = 4$
  • $\deg(C) = 4$
  • $\deg(D) = 4$
  • $\deg(E) = 4$

Euler Path: All vertices have even degrees, so an Euler path exists.

Euler Circuit: Since all vertices have even degrees, an Euler circuit also exists.

Summary of Results:

• $G_1$: Euler Path exists, but no Euler Circuit.
• $G_2$: Both Euler Path and Euler Circuit exist.
• $G_3$: Both Euler Path and Euler Circuit exist.

Would you like any more details or have questions?

Related Questions:

1. What is the difference between an Euler path and a Hamiltonian path?
2. Can a graph have more than one Euler circuit?
3. How can you quickly check if a graph has an Euler circuit?
4. What conditions guarantee the existence of an Euler path in a directed graph?
5. How does the concept of degree help in determining graph properties?

Tip:

A quick way to check for an Euler path or circuit is by counting the vertices with odd degrees—2 for a path, 0 for a circuit!