In the well-known article of Oystein Ore titled "A problem regarding the tracing of graphs", Elemente der Mathematik, 6 (1951), 49-53, he proves that an Euler graph $G$ is arbitrarily traceable from the vertex $a$ iff every cylcle of $G$ contains $a$. In his proof, he assumes, on the way of contradiction, that a cycle $C$ does not contain $a$. Consider the graph $G_1=G-C$ which is Eulerian, since the vertices of both $G$ and $C$ are even and hence the same holds for $G_1$. Now, he says that $a$ is contained in a "maximal" component $G^{(a)}$ of $G_1$. Let $G_2=G^{(a)}+C$. He claims that this $G_2$ is a connected Eulerian graph......
My questions:
what is meant by a "maximal" component of a graph?
why is $G_2$ a connected Eulerian graph? Thanks for any help!