On 1-factors with prescribed lengths in tournaments

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Title
On 1-factors with prescribed lengths in tournaments
Author(s)
Dong Yeap Kang; Kim J.
Publication Date
2020-03
Journal
JOURNAL OF COMBINATORIAL THEORY SERIES B, v.141, no., pp.31 - 71
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Abstract
© 2019 Elsevier Inc.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
URI
https://pr.ibs.re.kr/handle/8788114/6980
ISSN
0095-8956
Appears in Collections:
Pioneer Research Center for Mathematical and Computational Sciences(수리 및 계산과학 연구단) > Discrete Mathematics Group(이산 수학 그룹) > Journal Papers (저널논문)
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