On 1-factors with prescribed lengths in tournaments

Abstract

We prove that every strongly 10(50)t-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 Kuhn, Osthus, and Townsend. Indeed, we prove more results on partitioning tournaments. We prove that a strongly Omega(k(4)tq)-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 Omega(n). This result improves the earlier result of Kuhn, Osthus, and Townsend. We also prove that for a strongly Omega(t)-connected n-vertex tournament T and given 2t distinct vertices x(1), ... , x(t), y(1), ... , y(t) of T, we can find t vertex disjoint paths P-1, ... , P-t such that each path P-i connecting x(i) and y(i) has the prescribed length, provided that the prescribed lengths are Omega(n). For both results, the condition of connectivity being linear in t is best possible, and the condition of prescribed sizes being Omega(n) is also best possible.