To find an Euler circuit in the given graph (Figure 5-47), we need to ensure the graph meets the conditions for an Eulerian circuit:

Euler Circuit Conditions

1. Every vertex has an even degree – This ensures that you can enter and exit each vertex an equal number of times.
2. The graph is connected – Every vertex must be reachable from any other vertex.

Steps to Find an Euler Circuit

1. Verify the degree of each vertex:

   • Count the number of edges connected to each vertex.
   • If all vertices have an even degree, an Euler circuit exists.

2. Choose a starting vertex:

   • Any vertex can be the starting point if the graph meets the Euler circuit conditions.

3. Use Fleury’s Algorithm or Hierholzer’s Algorithm:

   • Fleury’s Algorithm: Avoid choosing edges that would disconnect the remaining graph.
   • Hierholzer’s Algorithm: Build a circuit and extend it until all edges are covered.

4. Label the edges in order of traversal:

   • Start from a vertex, traverse all edges exactly once, and return to the starting point.

Solution Approach

1. Identify an Euler circuit by systematically covering all edges.
2. Label the edges numerically in sequence.
3. Draw a modified graph with labeled edges.

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