K-ordered Hamiltonicity of Iterated Line Graphs

A graph G of order n is k-ordered hamiltonian, 2 ≤ k ≤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1, v2, . . . , vk in this order. In this paper, we generalize two well-known theorems of Chartrand on hamiltonicity of iterated line graphs to k-ordered hamiltonicity. We prove that if Ln(G) is k-ordered hamiltonian and n is sufficiently large, then Ln+1(G) is (k + 1)-ordered hamiltonian. Furthermore, for any connected graph G, which is not a path, cycle, or the claw K1,3, there exists an integer N ′ such that L N ′ +(k−3)(G) is k-ordered hamiltonian for k ≥ 3. c © 2008 Elsevier B.V. All rights reserved.