Is there a function $f : N\to N$ such that every $(k-1)$-connected graph with minimum degree at least $f(k)$ is at least $k$-connected?

Question

I tried to solve the following problem:

Is there a function $f: N \to N$ such that every $(k -1)$-connected graph with minimum degree, at least $f(k)$ is at least $k$-connected?

I have understood what k-connectivity means and also the meaning of the minimal degree. But I don't get to connect the two things. Does anyone have any tips on the solution for a function?

Thank you very much!

Answer 1

Fix $m$ and $n$ with $n\gg m$ and consider a "$K_m$ made of $K_n$'s", i.e., we have $m$ disjoint copies of $K_n$'s, pick a vertex from each of these, and add edges between any two of the $m$ picked vertices. The minimal degree of this graph is $n$, but the connectivity is determined by $m$.