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)). The minimum degree of a graph G is denoted by d(G).

A graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. 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 (d(Lⁱ(G))+1)-ordered. Since there is no graph H which is (d(H)+2)-ordered, the result is best possible.