Iterated line graphs are maximally ordered

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 tG such that for every t ≥ tG the t-iterated line graph of G, Lt (G), is (δ(Lt (G)) + 1)-ordered. Since there is no graph H which is (δ(H)+2)-ordered, the result is best possible. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 171–180, 2006