# On Min–Max Pair in Tournaments

On Min–Max Pair in Tournaments Graphs and Combinatorics https://doi.org/10.1007/s00373-018-1899-3 ORIGINAL PAPER Xiaoyun Lu Received: 14 March 2017 / Revised: 13 April 2018 © Springer Japan KK, part of Springer Nature 2018 Abstract Let T be a tournament of order n ≥ 3. A pair of distinct vertices x , y of T is called a min–max pair if one of x and y is of minimum out-degree, while the other is of maximum out-degree. Let xy be an arc such that x , y is a min–max pair. We call xy a min–max arc if x has minimum out-degree, and max–min arc otherwise. We prove that if yx is a min–max arc, then there exists a hamiltonian path from x to y;if xy is a max–min arc, then there exists a hamiltonian path from x to y with the exception of a few cases. Keywords Tournament · Hamiltonian path · min-max pair 1 Introduction A tournament T of order n is an orientation of a complete graph of order n, with vertex set V (T ) and arc set E (T ). Undeﬁned terminology can be found in [1]. If uv is an arc of T , then we say that u dominates v http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

# On Min–Max Pair in Tournaments

, Volume OnlineFirst – May 31, 2018
6 pages

/lp/springer_journal/on-min-max-pair-in-tournaments-NP50YtCeD1
Publisher
Springer Japan
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1899-3
Publisher site
See Article on Publisher Site

### Abstract

Graphs and Combinatorics https://doi.org/10.1007/s00373-018-1899-3 ORIGINAL PAPER Xiaoyun Lu Received: 14 March 2017 / Revised: 13 April 2018 © Springer Japan KK, part of Springer Nature 2018 Abstract Let T be a tournament of order n ≥ 3. A pair of distinct vertices x , y of T is called a min–max pair if one of x and y is of minimum out-degree, while the other is of maximum out-degree. Let xy be an arc such that x , y is a min–max pair. We call xy a min–max arc if x has minimum out-degree, and max–min arc otherwise. We prove that if yx is a min–max arc, then there exists a hamiltonian path from x to y;if xy is a max–min arc, then there exists a hamiltonian path from x to y with the exception of a few cases. Keywords Tournament · Hamiltonian path · min-max pair 1 Introduction A tournament T of order n is an orientation of a complete graph of order n, with vertex set V (T ) and arc set E (T ). Undeﬁned terminology can be found in [1]. If uv is an arc of T , then we say that u dominates v

### Journal

Graphs and CombinatoricsSpringer Journals

Published: May 31, 2018

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