On Sufficient Degree Conditions for a Graph to be $k$-linked
Combinatorics, Probability and Computing
An improved degree based condition for Hamiltonian cycles
Information Processing Letters
Minimum degree conditions for H-linked graphs
Discrete Applied Mathematics
Linkedness and ordered cycles in digraphs
Combinatorics, Probability and Computing
k-Ordered Hamilton cycles in digraphs
Journal of Combinatorial Theory Series B
Degree conditions on distance 2 vertices that imply k-ordered Hamiltonian
Discrete Applied Mathematics
A survey on Hamilton cycles in directed graphs
European Journal of Combinatorics
New Ore-Type Conditions for H-Linked Graphs
Journal of Graph Theory
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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. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = $\lceil{{n}\over{2}}\rceil + \lfloor{{k}\over{2}}\rfloor$ - 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k - 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) g(k, n) if 2k ≤ n ≤ 3k - 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999 An earlier version of this article was written while Sárközy was visiting MSRI Berkeley, as part of the Combinatorics Program.