在具有大最小半度的图中计算定向树

IF 1.2 1区 数学 Q1 MATHEMATICS
Felix Joos, Jonathan Schrodt
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If <em>G</em> is a digraph on <em>n</em> vertices with minimum semidegree <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi></math></span>, then <em>G</em> contains <em>T</em> as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of <em>T</em> the digraph <em>G</em> contains. 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引用次数: 0

摘要

设 T 是 n 个顶点上的定向树,其最大度最多为 eo(logn)。如果 G 是 n 个顶点上的数图,最小半度 δ0(G)≥(12+o(1))n,那么 G 包含作为生成树的 T,正如 Kathapurkar 和 Montgomery 最近证明的那样(事实上,他们只要求最大度为 o(n/logn))。这推广了 Komlós、Sárközy 和 Szemerédi 对图的相应结果。我们研究了数图 G 包含多少份 T 的自然问题。我们的主要结果表明,每个这样的 G 至少包含 T 的 |Aut(T)|-1(12-o(1))nn!这意味着无向情况下的类似结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Counting oriented trees in digraphs with large minimum semidegree

Let T be an oriented tree on n vertices with maximum degree at most eo(logn). If G is a digraph on n vertices with minimum semidegree δ0(G)(12+o(1))n, then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree o(n/logn)). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least |Aut(T)|1(12o(1))nn! copies of T, which is optimal. This implies the analogous result in the undirected case.

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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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