On 1-factors with prescribed lengths in tournaments

Abstract

We prove that every strongly 1050t-connected tournament contains all possible 1-factors with at most t components and this is best possible up to constant. In addition, we can ensure that each cycle in the 1-factor contains a prescribed vertex. This answers a question by Kühn, Osthus, and Townsend.Indeed, we prove more results on partitioning tournaments. We prove that a strongly Ω(k4tq)-connected tournament admits a vertex partition into t strongly k-connected tournaments with prescribed sizes such that each tournament contains q prescribed vertices, provided that the prescribed sizes are Ω(n). This result improves the earlier result of Kühn, Osthus, and Townsend. We also prove that for a strongly Ω(t)-connected n-vertex tournament T and given 2t distinct vertices x1,…,xt,y1,…,yt of T, we can find t vertex disjoint paths P1,…,Pt such that each path Pi connecting xi and yi has the prescribed length, provided that the prescribed lengths are Ω(n). For both results, the condition of connectivity being linear in t is best possible, and the condition of prescribed sizes being Ω(n) is also best possible.