Hamilton Transversals in Tournaments

IF 1 2区 数学 Q1 MATHEMATICS
Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo
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引用次数: 0

Abstract

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection \(\textbf{T}=(T_1,\dots ,T_m)\) of not-necessarily distinct tournaments on a common vertex set V, an m-edge directed graph \(\mathcal {D}\) with vertices in V is called a \(\textbf{T}\)-transversal if there exists a bijection \(\phi :E(\mathcal {D})\rightarrow [m]\) such that \(e\in E(T_{\phi (e)})\) for all \(e\in E(\mathcal {D})\). We prove that for sufficiently large m with \(m=|V|-1\), there exists a \(\textbf{T}\)-transversal Hamilton path. Moreover, if \(m=|V|\) and at least \(m-1\) of the tournaments \(T_1,\ldots ,T_m\) are assumed to be strongly connected, then there is a \(\textbf{T}\)-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub \(\textbf{H}\)-partition.

Abstract Image

锦标赛中的汉密尔顿横轴
众所周知,每个锦标赛都包含一条汉密尔顿路径,而每个强连接锦标赛都包含一个汉密尔顿循环。本文对这些经典结果进行了横向推广。对于共同顶点集 V 上不一定不同的锦标赛集合 (textbf{T}=(T_1,\dots ,T_m)\),如果存在双射 \(\phi :E(\mathcal{D})rightarrow[m]\),这样对于所有的E(\mathcal{D})\)来说,E(T_{\phi (e)})\(e\in E(T_{\phi (e)})\)都是横向的。我们证明,对于足够大的 m,且 \(m=|V|-1\),存在一条 \(\textbf{T}\)-transversal Hamilton 路径。此外,如果假定 \(m=|V|\)和至少 \(m-1\)个锦标赛 \(T_1,\ldots ,T_m\)是强连接的,那么就存在一个 \(\textbf{T}\)-transversal Hamilton 循环。在我们的证明中,我们使用了一种新颖的锦标赛分区方法,我们称之为 (textbf{H})分区。
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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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