2018/02/25 by Kang, Dong Yeap, Kim, Jaehoon
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1802.09049
Kühn, 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≥ 1, if a strongly Ck4t-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 Ω(n). This result improves the earlier result of Kühn, Osthus, and Townsend. Here, the condition of connectivity being linear in t is best possible, and the condition of prescribed size being Ω(n) is also best possible.