[Math] Semi Eulerian graphs

graph theory

I do not understand how it is possible to for a graph to be semi-Eulerian.

For a graph G to be Eulerian, it must be connected and every vertex must have even degree.

If something is semi-Eulerian then 2 vertices have odd degrees. But then G wont be connected.

Why wouldn't $G$ be connected? Look at the path graph:

$$ - - - - - -$$

Here is a $2$-regular graph that is not connected:

             o---o   o---o  
             |   |   |   |  
             o---o   o---o

Your reasoning about (b) and (c) is correct; (c) is the only one that must have an Euler circuit.

The claim you want to prove is false -- you've given a counterexample to it yourself.

So the answer to "How would you prove ..." is: That is impossible to prove, because it is not true.