-ordered hamiltonicity of iterated line graphs

A graph G of order n is k -ordered hamiltonian, 2 ≤ k ≤ n , if for every sequence v 1 , v 2 , … , v k of k distinct vertices of G , there exists a hamiltonian cycle that encounters v 1 , v 2 , … , v k 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 L n ( G ) is k -ordered hamiltonian and n is sufficiently large, then L n + 1 ( G ) is ( k + 1 ) -ordered hamiltonian. Furthermore, for any connected graph G , which is not a path, cycle, or the claw K 1 , 3 , there exists an integer N ′ such that L N ′ + ( k − 3 ) ( G ) is k -ordered hamiltonian for k ≥ 3 .