Is there a function f : N → N such that, for all k ∈ N, every graph of minimum degree at least f (k) is k-connected?

A graph H is called a subgraph of a graph G if V (H) ≤ V (G) and E(H) ≤ E(G); in this case we write HCG. Prove that every graph G with |E(G)| > 1 edges contains a subgraph H with minimum degree 8(H) > d(G)/2. (Hint: Construct a sequence of subgraphs G = Go 2 G₁ 2. as follows: if G; has a vertex vį, of degree da; (vi) ≤ d(Gi)/2, then obtain G₁+1 from Gį by deleting vį (which includes deleting all edges containing vį); if not, then terminate with H = Gį. Show that the resulting graph H has the desired properties.)

Is there a function f : N → N such that, for all k ∈ N, every graph of minimum degree at least f (k) is k-connected?