Abstract. By L(G) we denote the line graph of G. We set L⁰(G)=G. For i>0 the i-iterated line graph of G is Lⁱ(G)=L(L^i-1(G)). By d(G) we denote the minimum degree of G.

Let H be a graph with the minimum degree d(G)=d, d>=5, and let c=d.|_sqrt(d-1)_|, where sqrt means the square root. We prove that L³(H) contains K_c as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number i_G such that for every i>=i_G the i-iterated line graph of G is (1/2)d(Lⁱ(G))-linked. Since the degree of Lⁱ(G) is even, the result is best possible.