On 1-factors with prescribed lengths in tournaments.

*(English)*Zbl 1430.05048Summary: 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 D. Kühn et al. [Combinatorica 36, No. 4, 451–469 (2016; Zbl 1389.05058)].

Indeed, we prove more results on partitioning tournaments. We prove that a strongly \(\Omega(k^4 t q)\)-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 Kühn et al. [loc. cit.]. We also prove that for a strongly \(\Omega(t)\)-connected \(n\)-vertex tournament \(T\) and given \(2t\) distinct vertices \(x_1, \ldots, x_t, y_1, \ldots, y_t\) of \(T\), we can find \(t\) vertex disjoint paths \(P_1, \ldots, P_t\) 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.

Indeed, we prove more results on partitioning tournaments. We prove that a strongly \(\Omega(k^4 t q)\)-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 Kühn et al. [loc. cit.]. We also prove that for a strongly \(\Omega(t)\)-connected \(n\)-vertex tournament \(T\) and given \(2t\) distinct vertices \(x_1, \ldots, x_t, y_1, \ldots, y_t\) of \(T\), we can find \(t\) vertex disjoint paths \(P_1, \ldots, P_t\) 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.

##### MSC:

05C20 | Directed graphs (digraphs), tournaments |

05C38 | Paths and cycles |

05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |

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