Let $G_1$ and $G_2$ be $k$-connected graphs and let $v_1\in V(G_1)$ and $v_2\in V(G_2)$ be such that $\deg v_1=\deg v_2=k$. Form a new graph, $H$, by putting an $M$-matching of size $k$---conneect $N(v_1)$ with $N(v_2)$---between $G_1-v_1$ and $G_2-v_2$.
Show that $H$ is $k$-connected. I can easily show that $H$ is $(k-1)$-connected, but cannot show that it's $k$-connected. Any hints?
Show that for every pair of vertices $x,y$ there are $k$ internally disjoint $x,y$-paths, then use Menger's theorem.
Distinguish two cases.
If $x$ and $y$ are in different graphs, say $x\in G_1-v_1$ and $y\in G_2-v_2$, use an $x,N(v_1)$-fan and an $y,N(v_2)$-fan and glue them together using the matching.
If $x$ and $y$ are in the same graph, say $G_1$, then you can find $k$ internally disjoint $x,y$-paths in $G_1$. At most one of these paths contains $v_1$ and then it contains two neighbours of $v_1$. Now redirect the part of the path between these neighbours via the other graph.