k-ordered Hamiltonian graphs

A hamiltonian graph G of order n is k-ordered, 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. Theorems by Dirac and Ore, presenting sufficient conditions for a graph to be hamiltonian, are generalized to k-ordered hamiltonian graphs. The existence of k-ordered graphs with small maximum degree is investigated; in particular, a family of 4-regular 4-ordered graphs is described. A graph G of order n ≥ 3 is k-hamiltonian-connected, 2 ≤ k ≤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices, G contains a v1-vk hamiltonian path that encounters v1, v2, . . . , vk in this order. It is shown that for k ≥ 3, every (k+1)-hamiltonian-connected graph is k-ordered and a result of Ore on hamiltonian-connected graphs is generalized to k-hamiltonian-connected graphs. c © 1997 John Wiley & Sons, Inc. *Present Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307. †Present Address: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, NV 89154-4020. Journal of Graph Theory Vol. 24, No. 1, 45 57 (1997) c © 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/010045-13 46 JOURNAL OF GRAPH THEORY