Trees with many leaves in tournaments

IF 1.2 1区 数学 Q1 MATHEMATICS
Alistair Benford , Richard Montgomery
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引用次数: 0

Abstract

Sumner's universal tournament conjecture states that every (2n2)-vertex tournament should contain a copy of every n-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an (n+o(n))-vertex tournament may be sufficient.
In this paper, we give new results on these problems. Specifically, we show
  • i)
    for every α>0, there exists n0N such that, whenever nn0, every ((1+α)n+k)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves, and
  • ii)
    for every α>0, there exists c>0 and n0N such that, whenever nn0, every (1+α)n-vertex tournament contains a copy of every n-vertex oriented tree with maximum degree Δ(T)cn.
Our first result gives an asymptotic form of a conjecture by Havet and Thomassé, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree.
锦标赛中树叶繁茂的树木
萨姆纳的通用锦标赛猜想指出,每一个 (2n-2)- 顶点锦标赛都应该包含每一棵 n 个顶点的定向树的副本。如果我们知道一棵定向树的叶子数或它的最大度数,我们能否保证锦标赛中会有顶点数较少的定向树的副本呢?由于海格奎斯特(Häggkvist)和托马森(Thomason)(针对树叶数)以及库恩(Kühn)、迈克罗夫特(Mycroft)和奥斯特胡斯(Osthus)(针对最大度)所做的工作,我们知道在某些情况下可以改进萨姆纳猜想,实际上有时一个(n+o(n))顶点锦标赛可能就足够了。具体地说,我们证明i)对于每一个 α>0, 都存在 n0∈N 这样的情况:当 n⩾n0 时,每一个 ((1+α)n+k)-vertex tournament 都包含每一个有 k 个叶子的 n-vertex 定向树的副本;ii)对于每一个 α>;0,存在 c>0 和 n0∈N 这样的情况:当 n⩾n0 时,每一个 (1+α)n 顶点锦标赛都包含每一棵具有最大度 Δ(T)⩽cn 的 n 顶点定向树的副本。我们的第一个结果给出了 Havet 和 Thomassé 猜想的渐近形式,第二个结果改进了 Mycroft 和 Naia 的一个结果,该结果适用于最大度为多对数的树。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.70
自引率
14.30%
发文量
99
审稿时长
6-12 weeks
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
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