I've stumbled upon this question in Discrete math:

G is a simple graph with Vertices labeled {1,2,3,4,5,6,7}. It is given that G is Euler (that is, G has a Euler circuit) It is also given that there is no edge between vertices 1 and 2, 2 and 3, and 1 and 3. If we add G those edges, the graph we get is:

- Euler (that is, still contains a euler circuit)
- Not Euler, but contains a Eulerian trail.
- Not Euler, no Eulerian trail.
- Might be Euler, might not be. It depends on the original graph G
- It is not possible that G is Euler, because according to the data given, it is not a connected graph.

I think it's number 4, because I could draw the graph G in multiple ways according to the given Data, sometimes Getting a Euler circuit and sometimes not. However, I'm not sure about it. Mainly because I don't know whether it is Critical that a graph be connected for it to be Euler (they usually refer to connected graphs, but it's unclear whether it's critical for the definition of Euler).

Help would be much appreciated