## Abstract

A shortcut of a directed path v_{1}v_{2}⋯v_{n} is an edge v_{i}v_{j} with j>i+1. If j=i+2 the shortcut is a hop. If all hops are present, the path is called hop complete so the path and its hops form a square of a path. We prove that every tournament with n≥4 vertices has a Hamiltonian path with at least (4n−10)∕7 hops, and has a hop complete path of order at least n^{0.295}. A spanning binary tree of a tournament is a spanning shortcut tree if for every vertex of the tree, all its left descendants are in-neighbors and all its right descendants are out-neighbors. It is well-known that every tournament contains a spanning shortcut tree. The number of shortcuts of a shortcut tree is the number of shortcuts of its unique induced Hamiltonian path. Let t(n) denote the largest integer such that every tournament with n vertices has a spanning shortcut tree with at least t(n) shortcuts. We almost determine the asymptotic growth of t(n) as it is proved that [Formula presented].

Original language | English |
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Article number | 112168 |

Journal | Discrete Mathematics |

Volume | 344 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2021 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation (Grant No. 1082/16 ).

Publisher Copyright:

© 2020 Elsevier B.V.

## Keywords

- Hamiltonian path
- Shortcut
- Tournament

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics