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A new perspective from hypertournaments to tournaments

Jiangdong Ai, Qiming Dai, Qiwen Guo, Yingqi Hu, Changxin Wang

Abstract

A $k$-tournament $H$ on $n$ vertices is a pair $(V, A)$ for $2\leq k\leq n$, where $V(H)$ is a set of vertices, and $A(H)$ is a set of all possible $k$-tuples of vertices, such that for any $k$-subset $S$ of $V$, $A(H)$ contains exactly one of the $k!$ possible permutations of $S$. In this paper, we investigate the relationship between a hyperdigraph and its corresponding normal digraph. Particularly, drawing on a result from Gutin and Yeo, we establish an intrinsic relationship between a strong $k$-tournament and a strong tournament, which enables us to provide an alternative (more straightforward and concise) proof for some previously known results and get some new results.

A new perspective from hypertournaments to tournaments

Abstract

A -tournament on vertices is a pair for , where is a set of vertices, and is a set of all possible -tuples of vertices, such that for any -subset of , contains exactly one of the possible permutations of . In this paper, we investigate the relationship between a hyperdigraph and its corresponding normal digraph. Particularly, drawing on a result from Gutin and Yeo, we establish an intrinsic relationship between a strong -tournament and a strong tournament, which enables us to provide an alternative (more straightforward and concise) proof for some previously known results and get some new results.
Paper Structure (5 sections, 10 theorems, 12 equations)

This paper contains 5 sections, 10 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Strong $k$-tournament
  4. Concluding remarks
  5. Acknowledgement

Key Result

Theorem 1.1

Gutin1997 For $k\geq 3$, every $k$-tournament on $n\geq k+1$ vertices has a Hamiltonian path, and every strong $k$-tournament on $n\ge k+2$ vertices contains a Hamiltonian cycle.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 5 more