## Mathematics Faculty Books and Book Chapters

#### Chapter Title

Hankel Tournaments and Special Oriented Graphs

#### Book Title

Discrete Mathematics and Applications

#### Files

#### Document Type

Book Chapter

#### Editors

Andrei M. Raigorodskii and Michael Th. Rassias

#### Description

[Chapter Abstract]

A Hankel tournament *T* of order *n* (an *n* × *n* Hankel tournament matrix *T* = [*t*_{ij}]) is a tournament such that *i* → *j* an edge implies (*n* + 1 − *j*) → (*n* + 1 − *i*) is also an edge (*t*_{ij} = *t*_{n+1−j,n+1−i}) for all *i* and *j*. Hankel tournament matrices are (0, 1)-matrices which are combinatorially antisymmetric about the main diagonal and symmetric about the Hankel diagonal (the antidiagonal). Locally transitive tournaments are tournaments such that the in-neighborhood and the out-neighborhood of each vertex are transitive. Tournaments form a special class of oriented graphs. The score vectors of Hankel tournaments and of locally transitive tournaments have been characterized where each score vector of a locally transitive tournament is also a score vector of a Hankel tournament. In this paper we continue investigations into Hankel tournaments and locally transitive tournaments. We investigate Hankel cycles in Hankel tournaments and show in particular that a strongly connected Hankel tournament contains a Hankel Hamilton cycle and, in fact, is Hankel “even-pancyclic” or Hankel “odd-pancyclic.” We show that a Hankel score vector can be achieved by a Hankel “half-transitive” tournament, extending the corresponding result for score vectors of tournaments. We also consider some results on oriented graphs and the question of attainability of prescribed degrees by oriented graphs. Finally, we extend some results on 2-tournaments to Hankel 2-tournaments. In some instances we rely on the reader to extend arguments already in the literature. We illustrate our investigations with many examples.

#### ISBN

978-3-030-55857-4

#### Publication Date

2020

#### Publisher

Springer

#### Keywords

Tournament, Score vector, Locally transitive, Hankel tournament, Oriented graph, Cycle 2-Tournament

#### Disciplines

Mathematics | Physical Sciences and Mathematics

#### NSUWorks Citation

Brualdi, Richard and Lei Cao. (2020).
Hankel Tournaments and Special Oriented Graphs. In Andrei M. Raigorodskii and Michael Th. Rassias (Eds.), *Discrete Mathematics and Applications*
(109-152).

#### ORCID ID

0000-0001-7613-7191

#### ResearcherID

G-7341-2019

#### DOI

10.1007/978-3-030-55857-4

## Comments

Mathematics Subject Classifications

05C07 05C20 05C38 05C50 15B05