Consider the graph given above. Add an edge so the resulting graph has an Euler trail (without repeating an existing edge).
Now give an Euler trail through the graph with this new edge by listing the vertices in the order visited.
There can be at most two nodes of odd degree.
To add an edge to the graph so that the resulting graph has an Euler trail, we need to make sure that every vertex has an even degree except for two vertices that have an odd degree. An Euler trail is a path that visits every edge exactly once.
To find these vertices, we can determine the degree of each vertex in the given graph. The degree of a vertex is the number of edges connected to it.
By looking at the graph, we can count the degree of each vertex:
- Vertex A has degree 3
- Vertex B has degree 3
- Vertex C has degree 4
- Vertex D has degree 2
- Vertex E has degree 3
- Vertex F has degree 3
Notice that vertices C and D have an odd degree, while all the other vertices have an even degree. In order to add a new edge to form an Euler trail, we should choose one of these odd-degree vertices.
Let's say we decide to choose vertex C. We will need to find another vertex with an odd degree to connect to vertex C. Looking at the graph, vertex D is the only other vertex with an odd degree, so we will connect vertex C and vertex D with a new edge.
After adding the edge between C and D, the resulting graph will have the following degrees for each vertex:
- Vertex A has degree 3
- Vertex B has degree 3
- Vertex C has degree 5
- Vertex D has degree 3
- Vertex E has degree 3
- Vertex F has degree 3
Now, we need to find an Euler trail through the graph with this new edge. In order to find the trail, we can use the Hierholzer's algorithm:
1. Start at any vertex. Let's start at vertex A.
2. Follow any unused edge to an adjacent vertex.
3. Mark the chosen edge as used (remove it from the graph).
4. Repeat steps 2 and 3 until you return to the starting vertex.
5. If there are no unused edges, stop; otherwise, choose one of the remaining unused edges adjacent to the last visited vertex and repeat steps 2-5.
Here is an Euler trail through the graph with the new edge, starting at vertex A:
A -> B -> E -> D -> C -> F -> E -> B
By following this trail, we visit every edge exactly once and end up back at the starting vertex, which satisfies the definition of an Euler trail.