Linkedness and ordered cycles in digraphs

TL;DR

This paper proves that large digraphs with a minimum semi-degree above a certain threshold are guaranteed to be k-linked and k-ordered, confirming a longstanding conjecture and identifying optimal degree bounds.

Contribution

It establishes the exact minimum semi-degree needed for large digraphs to be k-linked and k-ordered, confirming a conjecture from 1990 and providing optimal bounds.

Findings

Minimum semi-degree at least n/2 + k - 1 ensures k-linkedness

Minimum semi-degree guarantees k-ordered cycles in large digraphs

Bound is proven to be best possible

Abstract

The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order.