On 1-factors with prescribed lengths in tournaments

TL;DR

This paper proves that strongly connected tournaments contain all 1-factors with up to t components, and extends to partitioning near-semicomplete digraphs into highly connected subgraphs with prescribed sizes, answering a question by Kuhn, Osthus, and Townsend.

Contribution

It establishes the existence of a constant C ensuring all 1-factors with up to t components are contained in strongly Ct-connected tournaments, and generalizes to partitioning near-semicomplete digraphs.

Findings

Strongly Ct-connected tournaments contain all 1-factors with up to t components.

Partitioning near-semicomplete digraphs into t strongly k-connected subgraphs is possible with prescribed sizes.

Conditions on connectivity and size are shown to be best possible.

Abstract

K\"uhn, Osthus, and Townsend asked whether there exists a constant $C$ such that every strongly $Ct$-connected tournament contains all possible $1$-factors with at most $t$ components. We answer this question in the affirmative. This is best possible up to constant. In addition, we can ensure that each cycle in the $1$-factor contains a prescribed vertex. Indeed, we derive this result from a more general result on partitioning digraphs which are close to semicomplete. More precisely, we prove that there exists a constant $C$ such that for any $k\geq 1$, if a strongly $Ck^4t$-connected digraph $D$ is close to semicomplete, then we can partition $D$ into $t$ strongly $k$-connected subgraphs with prescribed sizes, provided that the prescribed sizes are $\Omega(n)$. This result improves the earlier result of K\"uhn, Osthus, and Townsend. Here, the condition of connectivity being linear in $t$ is best possible, and the condition of prescribed size being $\Omega(n)$ is also best possible.