On 1-factors with prescribed lengths in tournaments
Dong Yeap Kang, Jaehoon Kim

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 such that every strongly -connected tournament contains all possible -factors with at most 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 -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 such that for any , if a strongly -connected digraph is close to semicomplete, then we can partition into strongly -connected subgraphs with prescribed sizes, provided that the prescribed sizes are . This result improves the earlier result of K\"uhn, Osthus, and Townsend. Here, the condition of connectivity being linear in…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Limits and Structures in Graph Theory
