k-Ordered Hamilton cycles in digraphs

TL;DR

This paper proves that large digraphs with a specific minimum semi-degree condition are guaranteed to contain a Hamilton cycle passing through any ordered sequence of k vertices, extending known undirected results to directed graphs.

Contribution

It establishes a tight minimum semi-degree condition for k-ordered Hamiltonicity in large digraphs, generalizing previous undirected graph results to directed graphs.

Findings

Minimum semi-degree at least (n+k)/2 -1 guarantees k-ordered Hamiltonicity.

Bound is proven to be optimal.

Extension of undirected results to directed graphs.

Abstract

Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, S\'ark\"ozy and Selkow.