given a disconnected graph $G$ with 2 components that in one of them all the vertices have even degrees (let's call it $Q$ and the other one $K$), prove that by adding one vertex (call it $s$) to G and edges from the added vertex to the original edeges we can get a simple graph $G'$ that has an eulerian path.
What I did so far:
I've looked at component K (not the one with all the vertices with even degrees):
if it has even number of odd-degree vertices let's put an edges between $s$ and every such vertex in $k$ , by doing that we get that all the vertices in $K$ are even-degree and $s$ has also even-degree. now let's put an edge between $s$ and some vertex in component $Q$ -let's call it $f$ - and now we have 2 vetices with odd degree-$s$ and $f$ and now we can say that $G'$ is connected and has an eulerian path.
The problem is the other case - when $K$ has odd number of odd-degree vertices. I don't know what to do in this case because the approach of the first case does not work here.